Question

Discrete Structures

1. Consider the following permutation function P, (a b c d e f g h i ghi e fac db) a. If P. is a permutation function on the set A, can you determine the set A with certainty (if so, write it below)? b. Represent P, as a set of ordered pairs. C. Find the digraph of P, d. Is P, a relation? e. Can a permutation function ever not be a relation? f. Is P, an equivalence relation? 8. Can a permutation function ever be an equivalence relation?

A={ab,bt,cq,qr,rc,ta,zz}

Answer 1

The solution to all the problems is pretty straight-forward.
An important point to note is that every function is a relation.


Solution Top-labe de f g h il Po hi e fa ed 6) [Page 1 (a) If P, is a permutation function on the set A, then P: A A is a bijection Thus, the set A can be determined with certainty and A = {a,b,c,d, erf, g, h, i} - (6) P {(ang),(b.n), (ei), cd,e), (ef), Cf, a), (8.c), (h.d),(6) (e) Digraph of Pass 09 s imbo R1-8)


( Page 2 (d) Yes P is a relation PC AXA and hence is a relation on A. Infact, every function is a relation and P, being a function is therefore a relation. No- Every permutation function being a function is a set of orderes pairs and hence in a relation always use a - (f) No Pi is not an equivalence relation. - It is becausse, Ca, a) e P, and this Pi is not reflexive. Thus, P, is not an equivalence relation.. (8) Yes. A permutation function is an equivalence e relation iff it is the identity function. Suppose that P is a permutation function that is an equivalence relation on a set s. Theng Ca.a) EP v AES. Since P is bijective, hence (a,b) EP e a=6 Thus, P = {canal: aES}. Thus P is the identity. Conversely, the identity function in an equivalence relation.

3. Ahab, bt, cq, ar, re, ta, 22} | Page 3 Po is a relation on A » XP y iff x's first letter is the same as y's second letter. (a) Yes. Po is a permutation function. (a 6) P₂ ta) (ore) P3 ax) (60) (ab) (ta) P₂ (6t) (C4) P (re) (22) 23 (22) (2x) P₂ (ex) (0) p. / ab ta се е х хe ta ?? ab re eq ar bt az