Second Term Lesson Note for Week Four
Class : SSS 1
Subject : Mathematics
Topic : Sets (meaning, types of set and terms used)
Duration : 80 Minutes
Period : Double Period
Reference Book :
New General Mathematics for Senior Secondary School, SSS 1.
Lagos State Unified schemes of work for Senior Secondary School, SSS 1 – 3.
Online Resources
Instructional Material : Chart showing the Set notation and their meanings.
Learning Objectives : By the end of the lesson learners will be able to :
i. Define Set.
ii. Identify the notation of Set and describe the.
iii. Mention the types of Set.
iv. Explain the Operation of Set.
Content :
Notation of Set, Types and Operation of Set.
Definition of Set.
A set is a well defined collection of objects or elements having some common characteristic or properties.
A set can be described by :
- Listing of its elements e.g. P= {b,a,d,c,a,b}
- Algebraic expressions e.g. B={X : 3 ≤ X ≤ 10}
- Statement form. e.g. A = {even numbers less than 10}
Giving a property that clearly defines its element
Notations used in set theory
Elements of a set : the members of a set are called elements e.g list the elements of set A = {even numbers less than 10}
A = {2, 4, 6, 8}
n(A) means number of elements contained in a set A. n(A) = 4
∈ means ‘is an element of or ‘belongs to’ e.g 6 ∈A
∉ means ‘is not an element of’ or ‘did not belong to’ e.g 5 ∉ A. defined in the above example.
(:) means such that e.g B={x : 3 ≤ x ≤ 10} means x is a member of B, such that x is a number that ranges from 3 to 10
Equal set: two sets are equal if they contain the same elements e.g If S = {a,d,c,b} and P= {b,a,d,c,a,b}, then S=P repeated elements are counted once
∅ or { } means empty set or null set i.e A set which has no element e.g
{secondary school student with age 3}
⊂ means subset. B is a subset of A, if all the elements of B are contained in A e.g If A = {1,2,3,4} and B = {1,2,3} then B is a subset of A i.e B ⊂A
∪ means union: all elements belonging to two or more given sets. A ∪ B means list all elements in A and B e.g.If A ={2,4,6,8,10} and B = {1,3,5,7,9} then A∪B = {1,2,3,4,5,6,7,8,9,10}
∩ means intersection i.e elements common to 2 or more sets e.g A ={1,2,3,4,5,6} and B = {1,3,5,7,9} then A∩B = {1,3,5}
µ and £ means universal set i.e a large set containing all the original given set i.e A set containing all elements in a given problem or situations under consideration
Complement of a set i.e A|A’| means ‘A complement’ and it is the set which contains elements that are not elements of set A but are in the universal set under consideration. E.g If E ={shoes and sock} and A={socks}, then |A’| ={shoes}
EVALUATION
1. State the elements in the given set below : Y= {Y : Y ∈ integer -4≤Y≤ 3}
2. Let £ = {x : 10<x< 20} P = {prime numbers} Q = {odd numbers}
Where P and Q are subsets of £.
(a) List all elements of set P
(b) What is n(P)?
(c) List all elements of set Q
(d) List the elements of P|
3. Make each of the following statements true by writing ∈ or ∉ in place of *
i.) 17 * {1,2,3,………7, 8,9}
ii.) 11 * {1,3,5,7…………. 19}
TYPES OF SETS
Universal set : A larger set containing all other sets under consideration i.e a set of students in a school
Finite set: is a set which contains a fixed number of elements. This means that a finite set has an end. E.g B = {1, 2, 3, 4, 5}
Infinite set: is a set which has unending number of elements or which has an infinite number of elements. An infinite set has no end of its elements. E.g. D = {5, 10, 15, 20, ….}
Subset of a Set : B is a subset of A if all elements of B are contained in A i.e. it is a smaller set contained in a larger or bigger set. E.g. If A = {1, 2, 3, 4, 5, 6} and B= {2, 3, 6} then B is a subset of A i.e. B ⊂ A
Empty set ∅ or { } : An empty set or null set contains no element
Disjoint set : if two sets have no elements in common, then they are said to be disjoint e.g If P = {2,5,7} and Q = {3,6,8} then P and Q are disjoint.
OPERATIONS OF SET
Intersection ∩ : the intersection of two sets A and B is the set containing the elements common to A and B e.g if A= {a,b,c,d,e} and B= {b,c,e,f}, then A ∩ B= {b,c,e}
Union ∪ : the union of A and B, A∪ B is a set which includes all elements of A and B e.g if A = {1,3} and B = {1,2,3,4,6}, then A∪B ={1,2,3,4,6}
Complement of a set: the complement of a set P, P| are elements of the universal set that that are not in P e.g if U = {1,2,3,4,5,6} P= {2,4,5,6}, then P|= {1,3}
Examples
Given that U = {a,b,c ,d,e,f}, P={b,d,e} Q= {b,c,e,f}
List the elements of
(a) P ∩ Q (b) P ∪ Q (c) (P ∩ Q)|
(d) (P ∪ Q)’ (e) P’ ∪ Q (f) Q’ ∩ P’
Solution :
P ∩ Q = {b,e}
P∪Q= {b, c, d, e, f}
Since (P ∩ Q ) = {b, e}
Then (P ∩ Q)| = {a, c, d, f}
Since (P∪Q) = {b, c, d, e, f}, then (P Ʋ Q)’ ={a}
P’ ∪Q
P’ = {a, c, f}
Q = {b, c, e, f}
Therefore P’ ∪ Q = {a, b, c, e, f}
Q’ = {a, d}
P’ = {b, d, e} = P’ ∩ Q’ = {d}
EVALUATION
Given that Universal set U= {1,2,3,4,5,6,7,8,9,10}, Set A= {2,4,6,8}, Set B= {1,2,5,9} and Set C= {2,3,9,10}
Find : (a) A∩B∩C (b) C’∩(A∩B)
(c) C∩(A∩B)’. (d) C’∪ (A∩B)
GENERAL EVALUATION
Given that U= {1,2,3…………19,20} and A ={1,2,4,9,19,20} B= {perfect square} C={factors of 24}. Where A,B, and C are subsets of universal set U
List all the elements of all the given sets
Find :
(i) n(A Ʋ B)|
(ii) n(A ƲB Ʋ C)
(iii) n(A|Ʋ B|∩ C)
Find (i) A∩B∩C (ii) AƲ(B ∩ C) (iii) (A|∩ B|)Ʋ C
List all the subsets of the following sets
A={Knife, Fork}
P={a, e, i}
READING ASSIGNMENT
NGM SSS1 page 71-72, exercise 5b and 5c.
WEEKEND ASSIGNMENT
If A={a, b, c} B={a, b, c, e} and C={a, b, c, d, e, f} find A∩B(AƲC) A.{a,b,c,d} B. {a,b,c,d,e} C.{a,b,d,d,e} D.{a,b,c}
If Q={0<x<30,x is a perfect square}, P={x÷1≤x≤10,x is an odd number} find Q∩P A.{1,3,9} B.{1,9,4} C.{1,9} D.{19,16,25}
Use the following information to answer questions 3 – 5
A,B and C are subsets of universal set U such that U={0,1,2,3……..11,12}, A={x:0<x<7}, B={4,6,8,10}, C={1<x<8}
Find (AƲC)| A{0,1,9} B.{2,3,4,5} C.{2,3,5,7} D.{0,1,2,9}
Find A|∩ B ∩C
A Ʋ B|∩ C A.{1,2,3,4,5,6,7} B.{2,3,5,7} C.{6,8,10,12} D.{4,5,7,9,11}
THEORY
The universal set U is the set of integers: A,B and C are subsets of U defined as follows
A= {….., -6,-4,-2,0,2,4,6…….}
B= {X: 0 <x < 9}
C= {X: -4 < x < 0}
Write down the set AI, where AI is the complement of A with respect to U
Find B∩C
Find the members of set BƲC, A∩B, and hence show that A∩(BƲC)=(A∩B)Ʋ(A∩C)
The universal set U is the set of all integers and the subsets P,Q,R of U are given by
P={X: X<0}, Q = {……,-5-,3,-1,1,3,5…….}, R= {X: -2<X<7}
Find Q∩ R
Find R| where R| is the complement of R with respect to U
Find P| ∩ R|
List the members of (P∩Q)