# investigation of all possible relations

I. (a) Show that. among the powers of 3 (1,3,9,27,81,243, . . .) there must be at least
two whose difference is divisible by 7. Hint: Group the numbers into 7 sets,
depending on what their remainder is when divided by 7.

Euzerwsmallest value of k that
371.31? a difference divisible
choice.
The next four questions will involve an investigation of all possible relations on the set
{1,2}.
2. There are four possible elmnmits in snvh a relation, namer a. A- l R Lb : 1 [32.6
2 R1. and d : â€˜2 R2. so any relation on that. set will be completely characterized by
the 16 possible. truth values for the propositions a. b,r-,(l.

(a) Let 1â€˜ denote whether a relation is reflexive. Show that r is equivalent to a A d.
Find a logical expreSSion in aâ€™ b. C. d
expreSSion in a, b. Câ€˜ d
3. Usin

g the 16 truth values for (a,b,c,d), fill in the 16
POSSlbllities of each â€˜bl . row truth table below for all the
Poss. c relation. Note. Donâ€™t mak
the table When youâ€˜r- l . â€“ a me read a mess; only fill in
0. sure you have it right.
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