SUBJECT: MATHEMATICS
CLASS: SS 1
DATE:
TERM: 2nd TERM
REFERENCE BOOK
- New General Mathematics SSS 1 M.F. Macrae et al
- WABP Essential Mathematics For Senior Secondary Schools 1 A.J.S Oluwasanmi
WEEK FOUR
TOPIC: IDEA OF SETS
CONTENT
- Notation of Set
- Types and Operation of Set.
Definition of Set
A set is a welldefined collection of objects or elements having some common characteristic or properties. A set can be described by
- Listing of its elements
- Giving a property that clearly defines its element
Notations used in set theory
- Elements of a set: the members of a set are called elementse.g list the elements of set
A = even numbers less than 10
- n(A) means number of elements contained in a set
- E means ‘is an element of or ‘belongs to’ e.g 6EA
- E means ‘is not an element of’ or‘did not belong to’ e.g 5 A defined in number 1 above
- (:) means such that e.g B={X : 3 ≤ X ≤ 10} means X is a member of B such that X is a number from 3 to 10
- Equal set: two sets are equal if they contain the same elements e.gIf S = {a,d,c,b} and P= {b,a,d,c,a,b}, then S=P repeated elements are counted once
- Ф 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.gIf A ={1,2,3,4} and B = {1,2,3} then B is a subset of A i.e B ⊂A
- U means union: all elements belonging to two or more given sets. A U 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 U B ={1,2,3,4,5,6,7,8,9,10}
- ∩ means intersection i.e elements common to 2 or more sets e.gA ={1,2,3,4,5,6} and B ={1,3,5,7,9} then A∩B = {1,3,5}
- Ʋ and E 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.gIf E ={shoes and sock} and A={socks}, then A| ={shoes}
EVALUATION
- State the elements in the given set below: Y= {Y: Y E integer -4≤Y≤ 3}
- Let E={x÷10
Where P and Q are subsets of E
- List all elements of set P (b) What is n(P)? (c) List all elements of set Q (d) List the elements of P|
- Make each of the following statements true by writing E or E in place of *
- 17 * 1,2,3,………7, 8,9 { }
- 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: B is a subset of A if all elements of B are contained in Ai.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 Ф 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 Ʋ: the union of A and B, A Ʋ 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 Ʋ 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
- P∩ Q (b) P Ʋ Q (c) (P ∩ Q)|
(d)(P Ʋ Q)| (e) P|Ʋ Q (f)Q|∩ P|
Solution
- P∩ Q = {b,e}
- P Ʋ Q= {b, c, d, e, f}
- Since (P ∩ Q ) = {b, e}
Then (P ∩ Q)| = {a, c, d, f}
- Sine (P Ʋ Q)= {b, c, d, e, f}, then (P Ʋ Q)| ={a}
- P|Ʋ Q
P| ={a, c, f}
Q={b, c, e, f}
Therefore P|Ʋ Q={a, b, c, e, f}
- Q| ={a, d}
P|={b, d, e} = P|∩ Q| = {d}
EVALUATION
Given that U= {1,2,3,4,5,6,7,8,9,10}, A= {2,4,6,8} B= {1,2,5,9} and C= {2,3,9,10}
Find: a) A∩B∩C (b) C|∩(A∩B) (c) C∩(A∩B)| (d) C|Ʋ(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
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- 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 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- 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)
