Exercise 1: Example : A-{ 1, 2), B= { a, b} A X B = { (1, a), (l, b), (2, a), (2

Question

Exercise 1: Example : A-{ 1, 2), B= { a, b} A X B = { (1, a), (l, b), (2, a), (2, b)) Solve the following: X= {a, c} and Y= {a,b,e,f) Write down the elements of (b) Yx X (d) What could you say about two sets A and BifAx B-Bx A? Exercise 2: Consider the following relation R on the set of S of TSU students, R={(x, y) | x and y have the same major). Prove that this relation is reflexive, symmetric and transitive. Exercise 3: For all sets and relations below, state and prove that the relation is 3) Anti-Symmetric 4) Transitive a) 1) Reflexive, 2) Symmetric b) Draw the digraph of each relation the set A-(2, 3, 4, 8,12,16), and the relation R- [(a,b)| a divides b) the set B- 3,7,9, 113 and the relation R((a.b) asb) - the set X-(1,2,3,4,5) and the relation R-(a,b)1sxs3&2sys4) Exercise 4 Let X = { San Francisco, Pittsburg, Chicago, San Diego, Philadelphia, Los Angeles) Define a relation R on X as x R y if x and y are in the same state (0) Show that R is an equivalence relation (b) List the equivalence classes of X

Explanation / Answer

This is long assignment to do at once. We are not allowed to answer long assignment. I have given you the answers for first two exercise. Please post the remaining questions seperately. Let me know if you have any doubt.

Exercise1:

a) X x Y = {(a,a),(a,b),(a,e),(a,f),(c,a),(c,b),(c,e),(c,f)}

b) Y x X = {(a,a),(a,c),(b,a),(b,c),(e,a),(e,c),(f,a),(f,c)}

c) X x X = {(a,a),(a,c),(c,a),(c,c)}

d) if AxB=BxA that means both set has same element.

Exercise 2:
1. a = a (reflexive property)
for both x any y, major is related to themselves. Hence, reflexive property is satisfied.

2. if a = b then b = a (symmetric property)

if x has major computer then y also has the major computer. Hence, it is symmetric i.e. x=y

3.if a = b and b = c then a = c (transitive property)

if x has major c and c is also taken by y that means x and y has the same major. Hence, it is also satisfies transitive property.

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