Back to: Further Mathematics SS1

**A. IDEA OF SET, NOTATIONS, APPLICATIONS.**

**Definitions**:

A set can be defined as a group or a collection of well defined objects or numbers e.g collection of books, cooking utensils.

A set is denoted by capital letters such as P, Q, and R e.t.c while small letters are used to denote the elements e.g. a, b, c

**Elements of a set:** These are the elements or members of a given set. The elements are separated by commas and enclosed by a curly bracket {}

e.g M ={ 1, 3 ,5, 7, 11}, 1 is an element of M.

Example: Write down the elements in each of the following sets.

A = {Odd numbers from 1 to 21}

F = {factors of 30}

M = {Multiples of 4 up to 40}

**Solution:**

A = { 1,3,5,7,9,11,13,15,17,19,21}

F = {1, 3, 5, 6, 10, 15, 30}

M = {4, 8, 12, 16, 20, 24, 28, 32, 36, 40}

** Cardinality of a set**: This is the number of elements in a set.

Example: Given that µ= {all the days of the week}, B= {all days of the week whose letter begin with s}

1. List all the elements of µ

2. List the members of B

3. What is n (µ) 4. What is n(µ) + n(B)

**Solution**:

1.µ = {Sunday, Monday, Tuesday, Wednesday, Thursday, Friday, Saturday}

2. B = {Sunday, Saturday}

3. n (µ) = 7

4. n (µ) + n(B) = 7 + 2= 9

**Set notation:** A set can be described algebraically using inequality and other symbols. E.g B = {x: -10≤x ≤ 3, x is an integer}

Example: List the members of the following sets

1. A= {x: 5< x< 8} 2. B= {x: 0≤ x≤ 5}

**Solution;**

1. A = {6, 7} 2. B = {0, 1, 2, 3, 4, 5}

**GENERAL/REVISION EVALUATION: **If µ= {all positive integers ≤ 30}, M= {all even number ≤ 20},

N = {all integers: 10≤ x≤ 30}

Find 1. n (µ) 2.n (N) 3. n (N) + n(s) 4. n (M) + n(N)

**B. Types of sets**:

Finite and Infinite set: Finite set is a set in which all its members can be listed.

Infinite set: An infinite set is a set in which all its members cannot be listed.

**Empty (Null) set**: A set without any member. It is usually denoted by { } or Ø.

**Subset and Supersets**: If we have 2 sets A and B such that all the elements in A is contained in B, then A is a subset of B. Subset is denoted by __C __e.g. A __C __B. If there is at least one element in set B but not in A, then B is a superset of A.

**Universal set:** This is a set that contains all the members under consideration for any given problem. It is denoted by µ or €.

**Complementary Set**: This is a set that contains the members in the universal set that are not in set A. It is denoted by A^{c } or A^{1}.

**Intersection of sets**: This is the set which consists all the common elements in a given two or more sets. It is denoted by n.

**Union of sets:** This is the set of all members that belong to A or to B or to both A and B. It is denoted by u.

Example: If the universal set µ= {x: 1≤ x ≤ 12} and its subsets D, F and G are given as follows. D = {x: 2<x<8}, F={x: 4≤ x≤ 10}, G={x: 1< x ≤ 4}

Find (a) D U F (b) D n F (c) G^{1 }(d) (D n G)^{1}

**Solution**:

µ = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}

D = {3, 4, 5, 6, 7}

F = {4, 5, 6, 7, 8, 9, 10}

G = {2, 3, 4}

(a) D U F ={ 3, 4, 5, 6, 7,8, 9,10}

(b) DnF = {4, 5, 6, 7}

(c) G^{1 } = { 1, 5, 6, 7, 8,9,10,11,12]

(d) (D n G)^{1 }

D n G = {3, 4}

(D n G)^{1 }= {1, 2, 5, 6, 7,8, 9,10,11,12}

**Relationship between union and intersection of sets**

n(A or B) = n( A) + n( B) –n( A and B)\

or n(AUB) = n(A) + n( B) – n(A n B)

Example:

If n(A)=23, n(B)= 12, n( AUB) = 35, find n(AnB) and comment on set A and B.

** Solution**

n(AUB) = n( A) + n( B) – n(AnB)

35 = 23 + 12 –n(AnB)

n(AnB) = 35- 35

n (AnB) = 0

Set A and B are disjoint.

**Evaluation:**

1. A and B are two sets. The number of elements in AUB is 49, the number in A is 22 and the number in B is

34.How many elements are in AnB?

2. The universal set µ ={ set of all integers}, p= {x:x≤ 2}, Q= { x: -7≤ x ≤15}R ={x: -2 ≤x ≤ 19}

Find 1. PnQ 2. P n (Q UR^{1})

**Venn diagrams:**

The Venn diagram is a geometric representation of sets using diagrams which shows different relationship between two or more sets. In order words, it is the diagrammatical representation of relationships between two or more sets. The operations of intersection, union and complementation of sets can be demonstrated by using Venn diagrams.

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