CSNB 143 Discrete Mathematical Structures Chapter 2 - Sets

Sets OBJECTIVES Student should be able to identify sets and its important components. Students should be able to apply set in daily lives. Students should know how to use set in its operations.

What, which, where, when 1. Basics of set (Clear / Not Clear ) 2.Terms used in set Equal sets (Clear / Not Clear) Empty (Clear / Not Clear) Disjoint / Joint (Clear / Not Clear) Finite / Infinite (Clear / Not Clear) Cardinality (Clear / Not Clear) Subset (Clear / Not Clear) Power set (Clear / Not Clear)

3. Operations on sets Union (Clear / Not Clear) Intersection (Clear / Not Clear) Complement (Clear / Not Clear) Symmetric Difference (Clear / Not Clear) 4. Venn Diagram Information Searching (Clear / Not Clear)

Set A collection of data or objects. Each entity is called element or member, defined by symbol . Order is not important. Repeated element is not important. One way to describe set is to listing all the elements, in curly bracket.

Ex 1: A = {1, 2, 3, 4, 5} B = {2, 3, 1, 4, 5) C = {1, 2, 1, 3, 4, 5} Thus we said, sets A and B are equal. A = B 1  A, 2  A, but 7  A How about A and C? How about B and C?

Ex 2: P = {p, q, r} Q = {p, q, r, s} R = {q, r, s} So, p  P, q  P, r  P, q  Q, r  R

Other way to describe set: A = {x| 1  x  5} A = {x| x is an integer from 1 to 5, both included} A = {x| x + 1 ; 0  x < 5} If the set has no element, it is called the empty set, denoted by {} or . Let D = {6, 7, 8} So, A and D are called Disjoint Sets. Why? What is the example of joined set?

A set A is called finite if it has n distinct elements, where n  N (nonnegative number). Ex 3: A = {x| 1  x  5} The number of its elements, n is called the cardinality of A, denoted by |A|= 5. A set that is not finite is called infinite. Ex 4: A = {x| x ≥ 1}

Subset If every element of A is also an element of B, that is, if whatever x  A then x  B, we say that A is a subset of B or that A is contained in B, written as A  B (some books use symbol ). Sets that all its elements are part or overall of other set. Ex 5: A = {1, 2, 3, 4, 5} B = {1, 3, 5} C = {1, 2, 4, 6}, Thus, B  A, but C  A B  A but A  B

Consider Ex 1. A = {1, 2, 3, 4, 5} B = {2, 3, 1, 4, 5) C = {1, 2, 1, 3, 4, 5} Is A  B? Is B  A? Is A  C? Is B  C?

Power Set If A is a set, then the set of all subsets of A is called the power set of A, denoted by P (A). A set that contains all its subset as its element. Ex 6: A = {1, 2} P (A) = {{1}, {2}, {1, 2}, } |P (A)| = 4

Operation on sets Union Let say A and B are sets. Their union is a set consisting of all elements that belong to A OR B and denoted by A  B. A  B = {x|x  A or x  B} Intersections Let say A and B are sets. Their intersection is a set consisting of all elements that belong to both A AND B and denoted by A  B. A  B = {x|x  A and x  B}

Operation on sets Complement Let say set U is a universal set. U – A is called the complement of A, denoted by A’ (some book use A) A’ = { x|x  A} If A and B are two sets, the complement of B with respect to A is a set that contain all elements that belong to A but not to B, denoted by A – B. Try Ex 5. Find A – B, A – C, C – A, C – B.

Symmetric Difference Let say A and B are two sets. Their symmetric difference is a set that contain all elements that belong to A OR B but not to both A and B, denoted by A  B. A  B = {x|(x  A and x  B) or (x  B and x  A)} Try Ex 2. Find P  R.

Venn Diagram Exercise : A  B A  B A  B  C A  B  C A – B B – A

Theorems Commutative A B = B  A A  B = B  A Associative A  (B  C) = (A  B)  C A  (B  C) = (A  B)  C Distributive A  (B  C) = (A  B)  (A  C) A  (B  C) = (A  B)  (A  C) Idempotent A  A = A A  A = A

Complement A’’ = A A  A’ = U A  A’ =  ’ = U U’ =  (A  B)’ = A’  B’ (A  B)’ = A’  B’ Universal Set A  U = U A  U = A Empty Set A   = A A   = 

2 disjoint sets |A  B| = |A| + |B| 2 joint sets |A  B| = |A| + |B| - |A  B| 3 disjoint sets |A  B  C| = |A| + |B| + |C| 3 joint sets |A  B  C| = |A| + |B| +|C| - |A  B| - |A  C| - |B  C| + |A  B  C|