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# Relations

Qu1) Is it true that &#961;(A&#8746;B)= &#961;(A) &#8746; &#961;(B)? justify your answer.

Qu2) Consider the function f:A&#8594;A defined by f(x)=x+1 and justify your answers.
a) For A=&#925; (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&#8594;R be given by x&#8594;3x-1 and g:R&#8594;R be given by x&#8594;x+1. Calculate (i) f&#959;g and ii) g&#959;f.
b) Prove that f&#959;g and g&#959;f are both invertible and describe their inverses.
c) Demonstrate that (f&#959;g)^-1 = g^-1 &#959; 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&#8712;Z
i) define aRb if and only if a^3&#8801;b^3 (mod 7)
Prove that R is an equivalence relation on Z.
ii) Define a=b if and only if a&#8801;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=&#12643;{(x,y)&#8712;A*A:xRy}&#12643;
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&#8712;{1,2,3,4,5} and a<b. Let relations R1 and R2 be defined on S as follows:
XR1Y &#8660; X&#8838;Y and XR2Y &#8660; X&#8745;Y &#8800;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.

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