CSNB 143 Discrete Mathematical Structures

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 setU 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 • A  B

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|

CSNB 143 Discrete Mathematical Structures