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Topic 2: Sets
 

 

1. SETS CONCEPT

1.1. Definition

A set is a group of objects. The objects in a set can be anything: students, names of countries, numbers, etc.  The objects in a set are called elements or members of the set. We write elements of the set in curly brackets, { }, sometimes called braces.

Set can be represented in two forms: roaster form or builder form. In roaster form the elements are enclosed by braces after separating them by commas. For example a set of natural numbers less than 6 is written as {1,2,3,4,5}. In this set there are five elements.

Here: set of natural numbers less than 6 is builder form or rule form and {1,2,3,4,5} is a roaster form or tabular form.

A set can also be described by using words written in curly brackets. For example, a set of positive integers less than ten may be presented as {positive integers less than ten}.

Sets are usually denoted by capital letters like A, B, C, D, … For example the set B of elements a and b is written as .

Note that in roaster form if the curly brackets are removed we lose the set (what remains is no longer a set).

Example

Use curly brackets to show the set of even numbers less than 10.

Solution

The even numbers less than 10 are 0,2,4,6 and 8, then the required set is {0, 2, 4, 6, 8}

Example

Use curly brackets to show the set of odd numbers less than 8.

Solution

The odd numbers less than 8 are 1, 3, 5 and 7, then the required set is {1, 3, 5, 7}

Example

Let S be the set of students by the names Jesse, Roy, Rooney and Hope. Write out that set using braces.

Solution



Finite set and infinite set

A set whose elements that can be listed down is said to be a finite set and a set whose elements that cannot be listed down is said to be an infinite set. A finite set has both ends closed but an infinite set has one end closed and the other open or both ends open.

Example

Use curly brackets to show the set of even numbers less than 13 and set of all even numbers.

Solution

Let A be the set of even numbers less than 13 and let B be the set of all even numbers.

Elements of A can be listed down. That is 2, 4, 6, 8, 10, 12; but elements of B cannot be listed down since we need to write 2, 4, 6, 8, 10, 12, 14, 16, … so we shall continue forever and ever. However in set theory, we solve this issue by adding three dots or dashes after the comma of the last member where we want to stop. Then,  and . Thus, A is a finite set and B is infinite set.

Example

Set of all prime numbers (prime number is a number that are divisible by 1 and itself only) is .

Set of counting numbers is

Set of integers (counting numbers with their negative, including zero) is

1.2. Belongs to and Cardinal number

Belongs and does not belong to

If set A contains an objects x as one of its members, then we write . This is read as x is a member of A or x belongs to A. If x is not one of the members of set A, then we write . This is read as x is not a member of A or x does not belong to A.

Example

Let . Use appropriate symbol to complete

1)                2)

Solution

1)                2)

Cardinal number of a set

Cardinal number of a set refers to the number of elements in the set. We write the cardinal of set A as . For example, given   then . If two sets have the same cardinal number, they said to be equivalent. If two sets contain exactly the same elements, they said to be equal or identical.

Example

Consider these sets:

The two sets are equivalent since they both have the same cardinal number, but they are also equal (or identical) sets because they contain exactly the same elements.

Note: When two sets are equal, as are F and G, we write . If not, we write

Example

For the given sets, give the cardinal number of each set and say whether they are equivalent or equal. A is the set of whole numbers less than 5 and set

Solution

Since   and , then .

The given sets are equivalents since they have the same cardinal number. But because their elements are not exactly the same.

Notice

- The empty set: Try to list the set of animals with six legs. Call the set E. This set has no element. An empty set is a set with no element. It is also called a null set or a void set. The empty set is denoted as but not both. For example, the set of natural numbers less that 0 is empty set since no natural number is less than zero.
- A singleton set: A singleton set (or unity set) is a set which has only one element. For example, the set of even numbers between 3 and 5 is a singleton set since this set has only one element which is 4.
- A pair set: A pair set is a set with exactly two elemements. For example the set of odd numbers between 2 and 6 is a pair set since it has two elements wich are 3 and 5.

1.3. Subsets

Sometimes when we work with sets, we pick some elements out of a set to form a new set. Consider the set: , we can pick the elements 2 and 3 to form set . This new set B is called the subset of set A.

Remark: If B is a subset of set A, all elements of B are contained in A.

The subsets relationship between sets is denoted by the symbol ⊂. Thus, in our case . If B is not a subset of set A, we write .

Number of subsets in a given set

Consider the set {2,3}. The sets {2}  and {3}  are subsets of {2,3}, but {2,3}  itself is also a subset and empty set is a subset of {2,3}. So the subsets of are {}, {1}, {2} and {2,3}.

Generally for a set with n elements, the number of subsets is given by . 

Example

Find the number of subsets in the set

Solution

In set  there are 5 elements, so the number of subsets is  

Example

Find the number of subsets in the set and hence list them.

Solution

In set  there are 3 elements, so the number of subsets is . Those subsets are and .

Note:

- Proper subset and improper subset: a proper subset of a set is a part of the set, but not all of the set, while an improper subset is all of the set or the set itself.

1.4. Venn diagram

Venn diagramm is a closed line where:

- each point in the inner part represents element of the set
- each point in the outside part represents an object which does not belong to the set
- each point element is represented once
- no element can be represented on the line.

Venn diagram for one set

Consider the elements 1, 2, 3, 7, 10 and set B. Let us represent on Venn diagram those elements if 1, 2 and 3 are the only elements of set B. We have the following diagram:

Here:

- The set B is represented by  the drawn circle, so element 1, 2 and 3 must be placed inside the circle since they are only elements of set B.

- Elements 7 and 10 are placed outside of the circle since they are not element of set B.

- The rectangular is used to represent the set of all elements under consideration. This set is called the universal set and is denoted by the letter .

- The set of elements 7 and 10 of universal set that are not element of set B is called the complement set of set B and is denoted by B'.

So here,

Venn diagram for two sets

The Venn diagram above divides the surface of the page into four regions: (1), (2), (3), (4).

- In region (1), we place the members common to both sets.
- In region (2), we place all the members in set A only.
- In region (3), we place all the members in set B only.
- In region (4), we place members belonging neither to A nor to B.

Venn diagram for three sets

The Venn diagram above divides the surface of the page into eight regions: (1), (2), (3), (4), (5), (6), (7), (8).

In region (1), we place the members common to all sets.
In region (2), we place all the members common to set A and B only.
In region (3), we place all the members common to set A and C only.
In region (4), we place all the members common to set B and C only.
In region (5), we place all the members in set A only.
In region (6), we place all the members in set B only.
In region (7), we place all the members in set C only.
In region (8), we place members belonging to none of the three sets.

Example

Represent using Venn diagram the following sets

1.

2.

Solution

Example

In a particular school it is compulsory that a student has to belong to at least one of the two societies; Arts and Science societies. Given that 600 students are members of both, 700 students belong to the Arts society and 800 to the Science society; determine the number of students in the school.

Solution

The given information can be represented on Venn diagram below

The number of students: 100+600+200=900 students

1.5. Intersection and union of sets

Intersection

The intersection of two sets A and B is another set whose elements are the elements common to both sets A and B. The common elements form a subset of sets A and B. We use the symbol ∩ for intersection.

Example

Let , then

Notice

1. Joint sets: Set A and B are said to be joint sets if their intersection is not empty set. That is if then A and B are joint sets.

2. Disjoint sets: For two sets A and B to be disjoint, no element of A is contained in B and no element of B is contained in A. That is, their intersection is empty set.

Examples

1. Let . A and B are joint sets since
2. Sets are disjoint sets since the intersection is the empty set.

Properties

If A, B and C are sets then

a) Commutative:
b) Associative:

Union of sets

The union set of two or more sets is another set whose elements form a list of all elements of the sets. We use the symbol ∪ for union of sets.

Example

Let . Then

Here, we note that a and b belongs to and so they are written once in the union set to avoid repetition.

Notice:

- If , then and hence
- If , then   and hence

Example

1.6. Set difference and symmetric difference

Sets difference

The set difference between set A and set B is the set of all members of set A that are not in set B. this is denoted by . We can write

Example

Given

a) Find
b) Show that

Solution

Symmetric difference

The symmetric difference between sets A and B is the set of all members of both sets A and B which do not belong to the intersection set. The symmetric difference is denoted by the symbol Δ. Thus, .

Example

Properties

2. RELATIONS

2.1. Ordered pair

An ordered pair consists of two elements separated by a comma. The two elements are between brackets and they are written in a specific order.

Recall that is a pair set whith exactly two elements.

Example

Determine x and y:

Solution

Example

Let us find the value of x if . Here

Cartesian product of two sets

The Cartesian product of two sets, say A and B, is denoted by . The Cartesian product is the set of all ordered pairs . That is .

If , then is denoted by . Thus,

Notice:

- If set A has p elements and set B has q elements then the Cartesian product of A and B has pq elements. Mathematically, .
-The Cartesian product of sets A and B can be described, in roaster form in three ways:
               - The sagittal diagram (or arrow diagram)
               - Listing all the orederd pairs
               - The Cartesian diagram

Example

Solution

Example

Solution

2.2. Relations

A relation is used to describe certain properties of things, mainly how long they are connected. A relation is a set of ordered pairs, whereby we separate sets of values, say A and B, and then join these two sets into one by matching member of set A to a member of set B.

For example, consider the sets . When we use the relation “is a square of” between these two sets we can list down the ordered pairs .

Using arrow diagram

Element in set B is the image of element in set A. Element in set A is the pre-image of element in set B. The set of all images under relation is called the range; the set of all pre-images under relation is called the domain of definition. If the range has fewer elements than those found in the image set, this image set in said to be the co-domain. Hence a range is a subset of a co-domain. In above relation, the set A is the domain; the set B is the range.

Example

Classification of relations

One to one relation

A relation is one to one if every element of the domain is mapped to its own single image.

Many to one relation

A relation is many to one if more than one preimage share the single image

One to many relation

A relation is one to many if there is an element in the domain mapped to more than one image.

Particular relations include functions. We will see functions in Topic 6.

2.3. Papygram

If relation R is defined from set A into itself, the sagittal diagram is called papygram. A papygram is obtained by linking the members in the set to each other by arrows.

Example

Let’s draw the papygram of the relation “... is less than or equal to ...” in the set .

The papygram is as follows:

Notice:

• The small loops in the papygram show that each element relates to itself
xRy means that x is related to y under the given relation R
xRx means that x is related to itself under this relation R

Properties of a relation in a set

• A relation R in set E is said to be reflexive if any element of E is related to itself, i.e xRx for any x in E.
• A relation R in set E is said to be symmetric if whenever xRy, we also have yRx
• A relation R in set E is said to be antisymmetric if we never have both xRy and yRx
• A relation R in set E is said to be transitive if for any two consecutive ordered pairs (x,y) and (y,z) verifying the relation, the resultant ordered pair (x,z) also verifies the relation, i.e if xRy and yRz then xRz, for any members x, y, z in E.

Example

Identify the property of each of the following relations

Solution

a) Reflexive
b) Reflexive, Symmetric
c) Reflexive, Antisymmetric

Notice:

Equivalence and ordering relations A relation R is said to be an equivalence relation if it is reflexive, symmetric and transitive. For an equivalence relation, the members in set are portioned into classes. A relation R is said to be an ordering relation if it is reflexive, antisymmetric and transitive.

Example

In this papygram below, the relation is equivalence

Set is divided into classes:

3. SETS OF NUMBERS

3.1. Natural numbers

Natural numbers (sometimes called the whole numbers) are those used for counting (as in “there are six coins on the table”) and ordering (as in “this is the third largest city in the country”). In common languages, the words used for counting are cardinal numbers and words used for ordering are ordinal numbers. The set of natural numbers is noted by .

There is no universal agreement about whether to include zero in the set of natural numbers. Some authors begin the natural number with zero and write , whereas others start with 1 and write ; including zero just supplies an identity element. If zero is included in set of natural numbers, the set becomes set of whole numbers. Thus, .

Natural numbers on number line:

Operations in

Addition

In the expression , c is the sum of a and b

Example

1) 1 + 3 = 4             2) 3 + 4 + 5 = 7 + 5 = 12

Subtraction

In the expression , c is the difference of a and b.

Example

1) 10 - 3 = 7               2) 30 - 4 - 2 = 26 - 2 = 24

Notice: Natural numbers are not closed under subtraction: . For example,

Multiplication

In the expression , c is the product of a and b.

Example

1)              2)

Division

In the expression , c is the quotient of a and b.

Natural numbers are not closed under division: . For example

Distributive property

This means that the multiplication is distributive over addition (or subtraction) in . The distributive property of multiplication over addition (or subtraction) is used to expand a product or factorize a sum.

Example

Expand

Solution

Notice:

To perform an operation containing addition, subtraction and products calculate the products and then calculate sums and differences.

3.2. Integers

The set of natural numbers and their negative, including zero, is the set of integers noted by . This means that the set of natural numbers is the subset of set of integers, i.e .

Positive integers .
Negative integers .

Thus,

Set of integers can be represented on a number line as follow

Notcie:

Properties of arithmetic operations in set of integers are the same as in set of natural numbers except for subtration for which the set of integers is closed under. That is, .

In addition, in set of integers there is additive inverse but no multiplicative inverses, except for -1 and 1. Additive inverse of a is -a, such that . For example, the additive inverse of 3 is -3.

3.3. Rational numbers

Any number that can be represented in the form (read as a over b), where a and b are integers and , is a rational number.

Since is a fraction notation for the quotient, the set of rational numbers is denoted by . In set notation: . Set of integers and set of natural numbers are subsets of the set of rational numbers. Thus, .

Example

The numbers are rational numbers.

Operations in set of rational numbers

Let consider the rational numbers and :

The set of rational numbers is closed under four operations (addition, subtraction, multiplication and division).

Example

Properties of operations in set of rational numbers

Given that

Notice:

A fraction can be proper, improper or mixed:

Proper fraction: is a fraction whose numerator is less than the denominator.

Example

Improper fraction: is a fraction with numerator either equal to or greater than the denominator.

Example

Mixed fraction: is a fraction that is a whole number and a proper fraction.

Example

A mixed fraction may always be converted into an improper fraction. To do this multiply the natural number by the denominator and add to the numerator. This new numerator over the denominator is the required fraction.

Example

Express as a mixed fraction.

Solution

Example

Express as a improper fraction.

Solution

Equivalent rational numbers

A given fraction number may be represented by different numbers. For example: are fractions representing the same fractional number. They can also be written as . Such a set of equal fractions is a set of equivalent fractions. In this case is said to be in its lowest term. Given a fraction number, we can find its equivalent fractional numbers by multiplying the numerator and the denominator by the same number different from zero.

Example

Find any two fractions equivalent to .

Solution

Cross product property

Two fractions are equivalent if and only if

Example

Show that   are equivalent rational numbers.

Solution

Example

Find a such that is equivalent to

Solution

3.4. Decimal and irrational numbers

Decimal system lets us write numbers of all types and sizes, using a clever symbol called the decimal point.

The decimal system has been extended to infinite decimals, for representing any real number, by using an infinite sequence of digits after the decimal separator. In this context, we can consider the following examples

Example

a) .
    These types of decimal are referred to as terminating decimals
b) .
    These are referred to as recurring decimals.
c) .
    These are referred to as non-recurring non-terminating decimals.

Notice:

An non terminating decimal represents a rational number if and only if it is a repeating decimal or is a terminating decimal.

Scientific notation

A number is expressed in scientific notation when there is one digit (not zero) before the decimal point. This number is multiplied by any power or exponent of ten, according to the digits we have to move across. We express a number in scientific notation as , where a is the number between 0 and 10 and n is an integer representing the number of digits we have moved across. Here decimal refers to base 10. The numbers like 345, 0.3456, 45.78, ... are expressed in decimal form.

Steps to follow:

1. Rewrite the number with a decimal point after the first figure on one side
2. Find out how many places to the left or right the decimal point has been moved. This number gives you the power of ten.

Example

Express the following in scientific notation

a) 1300                 b) 56700                        c) 9046000

Solution

Notice:

• The zeros after the decimal point are excluded
• When the number is in decimal notation the decimal point is at the end of the number.

Example

Express in scientific notation

a) 0.007              b) 43.56                  c) 8403.2                     d) 0.00056            e) 0.456

Solution

Example

Express in decimal notation

a)         b)

Solution

a)           b)

Converting from decimal form to fraction

As we saw, decimal numbers like 0.5, 0.25, 0.4, ... refer to as terminating decimal numbers and 0.333..., 0.272727..., refer to as recurring decimal numbers. A recurring decimal number in which a sequence of digits is repeated is called a periodic decimal and the sequence of digits that is repeated as a unit is called a period. The period of repeating decimal is indicated by placing a bar or a dot over the period.

Example

We can express a terminating decimal as a rational number. We need to know the number of decimal places is in the decimal number.

Example

Express 2.5 as a rational number

Solution

There is one decimal place, so we write . Simplifying we have

Example

Express 0.75 as a rational number

Solution

There are two decimal place, so we write . Simplifying we have

Also, we can express a recurring decimal as a rational number. We need to find a way of getting rid of the repeated decimal.

Example

Express as a rational number

Solution

Let

Multiply both sides by 10. That is

(because there is one period 3). Then we take off the recurring part:

Example

Express as a rational number.

Solution

Let

(Because there are two periods, we multiply by 100)

Example

Let us express 0.83333... as rational number

Solution

Let Because there is only one non-recurring decimal (8), we multiply by 10 in order to separate it from the recurring ones. That is,

Now

Irrational numbers

Some decimal numbers cannot be expressed as fractions. These numbers are called irrational numbers. These are numbers that are decimal and not terminate or repeat but go one without end. The set of irrational numbers is denoted by (means the complement of rational numbers, ).

Example

1.010203040506..., 0.123456789... are irrational numbers. is another example of an irrational number, . The fraction is often used for the value of .

In general, if a number is not a perfect square (if it is not a product of two similar numbers) its square root is irrational. These square roots are called surds. For example: are surds. We will see more on surds later.

Both rational and irrational numbers are subsets of decimal numbers.

3.5. Real numbers

3.5.1. Definition

The set of real numbers is the set of all rational numbers together with all irrational numbers. It is denoted by . Thus, . Using the symbol of subsets we may write .

When an origin, a scale and a direction are chosen on a line, then to any point of that line corresponds a unique real number. When all real numbers are placed on that line, they cover the line completely with no gaps. That line is called number line and is shown below

Properties of the arithmetic operations in set of real numbers

3.5.2. Powers and radicals

Powers

We call nth power of a real number b that we note , the product of n factors of b. that is

n is an exponent and b is the base.

Example

Note that and is not defined

Example

Simplify

1)             2)

Solution

Radicals

The nth root of a real number b is denoted by where

Notice:

Example

Remark:

If n=2, we say square root and  is written as . Here b must be a positive real number or zero.
If n=3, we say cube root noted . Here b can be any real number.
If n=4, we say fourth root noted . Here b must be a positive real number or zero.
and so on.

In general, if the indice is even, the radicand must be postive real number or zero and if the indice is odd, the radicand can be any real number.

Example

is not defined in set of real numbers but

Properties

Similar radicals

Similar radicals are the radicals with the same indices and same bases. Simplification of sum (or difference) of radicals is possible when we have similar radicals.

Example

Multiplication of radicals

When multiplying radicals we may need to reduce to the same indice

Example

Suppose that we need to reduce to the same indice the radicals and .
The two radicals have indices 4 and 3 respectively. When reducing them to the same indice, the resulting radical must be the product of the indices. That is the resulting indice is .
We do that as follow

Rationalizing

Rationalizing is to convert a fraction with an irrational denominator to a fraction with rational denominator.

To do this, if the denominator involves radicals we multiply the numerator and denominator by the conjugate of the denominator. We will the case where the denominator contains square roots only.

Note the following:

Example

a)

b)

c)

                           

                           

Square roots

The square root of a given number is the number which when multiplied by itself, will produce the given number. For example the square root of 4 is 2 because 2 multiplied by 2 gives 4.

To find square root of a given number we may use inspection method, prime factor method or estimation method.

Inspection method

To find the square root, of a given real number, by inspection method we find another real number such the product of that real number by itself will produce the given real number.

Example

By inspection, find the square roots of each of the following

1) 64                                2) 144                                     3) 225

Solution

Prime factor method

When a number is written in prime factor form (i.e. the product of prime numbers), its square root can be found by dividing each exponent by 2.

Example

Find, by prime factor method, the square root of

1) 64                          2) 144                                3) 225

Solution

Estimation method

If a number does not have two equal factors, we can estimate the square root and check the estimate by squaring it.

Example

Use the estimation method to find the square root of 10.

Solution

We take two consecutive numbers such that 10 will lie between their squares. Take 3 and 4.

Because and , the square root of 10 is greater than 3 and less than 4.

Try , this is less than 10 this is greater than 10
      , this is greater than 10

Then square root of 10 is between 3.1 and 3.2. Since these two values are consecutive we take one of them for which the square is closer to 10 than other. We see that 10.24 is closer to 10 than 9.24

10.24 and 10, the difference is 0.24
9.24 and 10, the difference is 0.39.

Then we take the value whose square is 10.24. Thus, . We say that is a surd.

Square root by general method

To find the square root of a positive integer, say 103041, proceed as follow:

Explanation:

a) Mark off the digits from the right in groups of two (the last may consist of only one digit)
b) Find the greatest integer whose square is not greater than 10, in this example take 3
c) Put 3 above 10, as shown and also as the divisor
d) Take the square of 3 (i.e 9) and subtract it from 10 to get 1, and lower the next group of two digits, i.e 30, to get 130.
e) Double the quotient 3 on top, to get 6 as the first part of the next divisor. The next part of the divisor is the digit such that when the digit is added to 6 and the result multiplied by the digit, the product is not greater than 130. In our example, the digit is 2. We obtain the divisor 62.
f) Put 2 above the second group (30), as shown. Multiply the divisor 62 by 2 then subtract the product from 130 to get 6. Lower the last group of digits (41) to get 641.
g) Double the quotient 32 on top to get 64 as the first part of the divisor. Find a digit such that when it is added to 64, and the resulting divisor multiplied by the digit, the product is not greater than 641. In our example, the digit is 1.
h) Put 1 above the last group (41) to get 321 as shown in the example, and obtain the divisor 641, then multiply it by 1, subtract the product (641) from 641.
i) The remainder is 0. Therefore .

3.5.3. Absolute value

Absolute value of a number is the number of units it is from 0 on a number line. The symbol | | is used to denote the absolute value.

Example

6 is at 6 units from zero, thus the absolute value of 6 is 6. Or . Also -6 is at 6 units from zero, thus the absolute value of -6 is 6. Or . So since -6 and 6 are on equal distance from zero on number line.

Notice:

• The absolute value of zero is zero
• The absolute value of a non-zero real number is a positive real number.
• Given that . This will be used to solve equations or inequalities involving absolute value in next topic, Equations and Inequalities.

Example

Simplify

a)             b)              c)

Solution

3.5.4. Logarithms

a being a positive real number different from 1, we call logarithm of a positive real number x with base a the number noted ad defined by

where ln is natural logarithm.

In case , we simply write (decimal logarithm)
In case , we simply write (natural logarithm). We will see this in Logarithmic functions.

Properties

Also,


These relation will be used to solve logarithmic equations and inequalities.

Example

Write as one logarithm:

Solution

Example

Known that , calculate

Solution

Change of base

Suppose that we need to change in base b.

Thus, 

Example

Change in base 3

Solution

3.5.5. Polynomials

Definitions

In algebraic expression the parts that are added (or subtracted) are called terms. The numerical factor of a term is called a coefficient. Like terms have the same variables raised to the same powers and only their coefficient may differ. For example,   are like terms.

A monomial is an expression that is either a real number or a variable with natural exponent. For example  are monomials but   are not monomials.

A polynomial is a sum of monomials. For example, is a polynomial. A polynomial with two terms is called a binomial and a polynomial with three terms is called a trinomial.

The highest exponent of any term in a polynomial determines the degree of the polynomial. For example, the polynomial   have degree 5. A one variable polynomial is in standard form if the terms in the polynomial are in descending order by degree, and no two terms are alike.

When simplifying polynomial we combine the like terms.

Example

Simplify

1)                                         2)

Solution

1)                 2)

Numerical calculation

To obtain the numerical value of the expression that contains one or more unknown, substitute numbers for the unknown.

Example

Evaluate when

Solution

Example

Find the numerical value of for x=3

Solution

Operations on polynomials

Addition

Addition of polynomials is performed by grouping the like terms and simplifying.

Example

Given that and , find and

Solution

Multiplication

When multiplying two polynomials we use distributive property of multiplication over addition. The process is called expansion where we transform product to sum.

Example

1. Given that and , find the product

2. Expand the expression and simplify

Solution

1.

2.

Notice: Algebraic identities given below are sometime used in expansion of similar expression

Example

Expand

1.           2.

Solution

Factorization of polynomials

Factorization is the reverse of expansion where we transform sum to product. When factorizing an expression, look for common factors first.

Example

Factorize the following expressions

Solution

Use of algebraic identities

Some time we can use the algebraic identities to factorize a quadratic expression

Example

Put the following expression in factor form

1)                    2)

Solution

1)

This is in the form . Here .

Thus,

2)

This is in the form . Here .

Thus,

Factorization of quadratic expressions

A quadratic expression has the form . Factor form of this expressions is so that . Here the sum and the product .

Example

Factorize

1)          2)              3)

Solution

1) We need such that

. Then,

2) We need such that

. Then,

3) We need such that .
Here, it is not easy to find with the same method as in examples 1 and 2. What we need is to multiply the coefficient of by the constant 5. That is . Find negative factors of the product 15 whose sum is -16

Rewrite the term using the factors we found.

Discriminant method

When factorizing the expression . We find

If Δ>0, exist and the factor form is
Here

If Δ=0, are equal and the factor form is
Here

If Δ<0, do not exist and there is no factor form.

Example

1)                         2)                        3)

Solution

Division of polynomials

a) Denominator is a monomial

This is similar to numerical division

Example

1) Simplify

2) Divide by

3) Simplify

Solution

b) Denominator is a polynomial

Method 1: Long division

To find the quotient and the remainder, we can proceed as shown in the following examples

Example

Divide by

Solution

Example

Given that , divide

Solution

Method 2: Synthetic division

To use synthetic division, we first need to know the remainder theorem.

The remainder theorem states that the remainder of the division of polynomial by is . Hence, is divisible by if and only if . In this case, is a factor of .

Example

Use synthetic division to find the quotient and the remainder of the division

Solution

Example

Use the remainder theorem to show that is divisible by . Use synthetic division to find the quotient

Solution

3.5.6. Groups, Rings and Fields

Binary Operation

A binary operation in set E is any function mapping a pair taken from a set onto a single element from that set. For example 3 and 4 are taken from set of integers and the operation (addition) maps them to 7 which is also an elements of set of integers.

Example

a) Multiplication is a binary operation in set of natural numbers since the product of any two natural numbers is another natural number.
b) Subtraction is a binary operation in set of integers since the difference between any two integers is another integer.

An external binary operation is a binary operation from to S. This differ from a binary operation in the strict sense in that K need not be S; its elements come from outside. For example, the multiplication of real numbers by vectors is an external binary operation.

Properties

a) Closure property

The operation * is said to verify the closure property in set E, or set E is closed under operation *, if and only if, for any members x and y in E,   is always in E.

Example

a) The addition and multiplication verify the closure property in set of integers (or set of integers is closed under addition and multiplication). In fact, for .
b) The subtraction does not verify the closure property in set of natural numbers. For example .

b) Commutative property

Operation * is said to verify the commutative property in set E, or * is commutative in E, if and only if, for any members x, and y in E, .

Example

Addition and multiplication are commutative in

Subtraction is not commutative in . For example

c) Associative property

Operation * is said to verify the associative property in set E, or * is associative in E, if and only if, for any members x, y and z in E, .

Example

Addition and multiplication are associative in

d) Identity property

Operation * is said to verify the identity property in a non empty set E if there exists an element, say e, such that for any element x in E; . e is said to be an identity element for operation * in E.

0 is the identity element for addition in ;
1 is the identity element for multiplication in ;

Example

Find the identity element for operation T defined in by

Solution

Let e be the identity element. Then

Similarly,

The identity element is -3

e) Inverse property

Operation * is said to verify the inverse property in a non empty set E, for any element x in E, there exists an element x' in E such that: where e is the identity element for operation * in E. x' is said to be the inverse of x under operation *.

. x' is said to be the opposite of x (or additive inverse of x) .

. x' is said to be the multiplicative inverse of x.

Example

a) . Then -3 is the additive inverse (opposite) of 3.
b) . Then is the multiplicative inverse of 3.

Example

Find the inverse of 1 under the operation T defined by in

Solution

We saw that for this operation the identity element is -3 Let x be the inverse of 1. Then

The inverse of 1 is -7

f) Distributive property

Operation * is said to be distributive over operation T in set E if and only if for any members x, y and z in E,

: * is left distributive over T
: * is right distributive over T
Multiplication is distributive over addition in .

Example

Expand in the following expressions:

a)                        b)

Solution

a)             b)

A set equipped with a single binary operation is called magma.

Types of magmas

Unital magma is magma with identity element.
Semigroup (or associative magma) is magma where operation is associative.
Monoid (groupoid) is semigroup with identity element.
Group is monoid with inverse element.
Abelian (commutative) group is group where the operation is commutative.

Cayley table of a binary operation

Cayley (or composition) table of a binary operation is the tabular form of all possible outcomes of binary operation * on a finite set G written in a square array having n+1 rows and columns, where n is the total number of elements of G.

Example

Let A={1,2,3,4,5} and T be a binary operation on A defined by . Construct the Cayley table for the operation T. Use the table to solve    

a)               b)

Solution

The Cayley table is

Algebraic structure

Group concept

Let G be a non-empty set and * an operation defined in G. The ordered pair (G,*) is said to be a group if and only if:

i) G is closed under operation *
ii) * is associative
iii) * verifies the identity property in G
iv) * verifies the inverse property in G.
If * is commutative, the ordered pair (G,*) is said to be an abelian or commutative group.

If at least one of the four properties is not satisfied, then is (G,*) not a group.

Example

is an abelian group. In fact,

i) Closure:
ii) Associative:
iii) 0 is identity element:
iv) –x is additive inverse of x:

Since all four properties are satisfied, we conclude that is a group.

Moreover, addition is commutative in . i.e, . Then is an abelian group.

Ring concept

Let R be a non empty set closed under operations * and T. If

i) (R,*) is an abelian group
ii) R is closed under operation T and operation T is associative
iii) Operation T is distributive over *
Then, (R,*) is a ring

Example

is a ring. In fact,

i) is an abelian group, from previous example.
ii) is closed under multiplication and associative:
iii) Multiplication is distributive over addition in :

Field concept

The ordered triple (F,*,T) is said to be a field if

i) (F,*) is an abelian group
ii) (F- {e},T) is an abelian group
iii) T is distributive over *

Where (F- {e},T) is the set of all elements in F except the identity element e for the operation *

Example

is a field. In fact,

i) is an abelian group
ii) is an abelian group
iii) Multiplication is distributive over addition

4. APLICATIONS

Venn diagram in real life problems

Sets can be used to solve word problems. To solve word problem, we first need to represent the problem using sets and then solve the problem and interpret the results.

Example

Out of forty students, 14 are taking English Composition and 29 are taking Chemistry.

  • If five students are in both classes, how many students are in neither class?
  • How many are in either class?

Solution

a) 2 students are in neither class

b) 9+5+24 = 38 students are in either English or Chemistry.

References

  1. E. Ngezahayo & P. Icyingeneye. Advanced Mathematics for Rwanda Schools Learner's Book 4, Fountain Publishers Ltd, 2016.
  2. E. Ngezahayo & P. Icyingeneye. Advanced Mathematics for Rwanda Schools Learner's Book 5, Fountain Publishers Ltd, 2016.

 

 

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