CSNB143 – Discrete Structure Topic 1 - Set

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.

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

Topic 1 - Sets Example : Work this out : P = {p, q, r}Q = {p, q, r, s}R = {q, r, s} StatementTrue or False? q  Q, T/F r  R T/F s  P, T/F p  P T/F StatementTrue or False? q  Q, T/F p  R T/F s  P, T/F s  Q T/F

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 . LetD = {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}

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 Is A  B?Is B  A?Is A  C?Is B  C?

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

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}

Topic 1 - Sets Operations 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. 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

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