# Relations

Qu1) Is it true that ρ(AUB)= ρ(A) U ρ(B)? justify your answer.

Qu2) Consider the function f:A→A defined by f(x)=x+1 and justify your answers.

a) For A=ν (integers) is f onto?

b) For A=R(real number) is f injective?

c) For A=Q (rationals) is f onto?

d) For A=Z(all integers) is f a bijection?

Q3)

a) Let f : R→R be given by x→3x-1 and g:R→R be given by x→x+1. Calculate (i) fοg and ii) gοf.

b) Prove that fοg and gοf are both invertible and describe their inverses.

c) Demonstrate that (fοg)^-1 = g^-1 ο f^-1

Q4) a) Let A= {(1,3),(2,4),(-4,-8),(3,9),(1,5),(3,6)}. Define a relation R on A as follows: (a,b)R(c,d) if ad=bc. List the equivalence classes of R.

b) Let a,b∈Z

i) define aRb if and only if a^3≡b^3 (mod 7)

Prove that R is an equivalence relation on Z.

ii) Define a=b if and only if a≡b (mod 7). What are the equivalence classes for =?

Q5) On the set {a,b,c} consider the following relations.

a) R1={(a,a), (a,b), (a,c)}

b) R2={(a,a), (b,b), (c,c)}

c) R3={(a,a), (a,b), (b,a)}

Fore each of these relations, decide if it is reflexive, symmetric, transitive, antisymmetric. Justify your answers.

Q6) Let A be a set with 5 elements and let n=ㅣ{(x,y)∈A*A:xRy}ㅣ

a) If R is a partial ordering what is the minimum value of n?

b) If R is a total ordering what is the value of n?

c) If R is a partial ordering what is the maximum value of n?

Q7) Let S be the set of all intervals [a,b] such that a,b∈{1,2,3,4,5} and a<b. Let relations R1 and R2 be defined on S as follows:

XR1Y ⇔ X⊆Y and XR2Y ⇔ X∩Y ≠Ø

a) Determine whether or not R1 is a partial ordering on S and whether or not R2 is a partial ordering on S.

b) If (S,R1) and/or (S,R2) is a poset, then draw its Hasse diagram and decide whether it is a lattice.

#### Solution Summary

This is a set of questions regarding relations and Hasse diagrams.