CSNB143 – Discrete Structure

1. CSNB143 – Discrete Structure Topic 1 - Set

2. Topic 1 - Sets Learning Outcomes • 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.

3. Topic 1- Set Introduction • 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 list all the elements, in curly brackets. 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

4. Topic 1 - Sets Example : Work this out :

5. Topic 1 - Sets • 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} , A and D are called then Disjoint Sets. Why? What is the example of joined set? • Set A is called finite if it has n distinct elements, where n N (nonnegative number). Example : R = {x| 1  x  5} • The number of its elements, n is called the cardinality of R, denoted by |R|= 5. • A set that is not finite is called infinite. Example : C = {x| x ≥ 1}

6. Topic 1 - Sets Subsets • If every element of A is also an element of 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. • Example : 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 • Work this out

7. Topic 1 - Sets 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. • Example: A = {1, 2} P (A) = {{1}, {2}, {1, 2}, } P (A)| = 4

8. Topic 1 - Sets Operations 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}

9. Topic 1 - Sets Operations 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. 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)} Find P R

10. Topic 1 - Sets Venn Diagram • Diagram that is used to show the relations between sets. • Example : Given set A = {1, 2, 3, 4} and B = {1, 3, 5, 7, 9} • Show using Venn diagram: a) A  B and A  B b) A  B = A B – (A  B) c) A’ and B’ d) A – B and B - A