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I. (a) Show that. among the powers of 3 (1,3,9,27,81,243, . . .) there must be at leasttwo 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 : ‘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 1‘ denote whether a relation is reflexive. Show that r is equivalent to a A d.

Find a logical expreSSion in a’ b. C. d

expreSSion in a, b. C‘ d

3. Usin

g the 16 truth values for (a,b,c,d), fill in the 16

POSSlbllities of each ‘bl . row truth table below for all the

Poss. c relation. Note. Don’t mak

the table When you‘r- l . – a me read a mess; only fill in

0. sure you have it right.

See next page

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March 21, 2017

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I. (a) Show that. among the powers of 3 (1,3,9,27,81,243, . . .) there must be at leasttwo 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 : ‘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 1‘ denote whether a relation is reflexive. Show that r is equivalent to a A d.

Find a logical expreSSion in a’ b. C. d

expreSSion in a, b. C‘ d

3. Usin

g the 16 truth values for (a,b,c,d), fill in the 16

POSSlbllities of each ‘bl . row truth table below for all the

Poss. c relation. Note. Don’t mak

the table When you‘r- l . – a me read a mess; only fill in

0. sure you have it right.

See next page

I. (a) Show that. among the powers of 3 (1,3,9,27,81,243, . . .) there must be at leasttwo 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 : ‘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 1‘ denote whether a relation is reflexive. Show that r is equivalent to a A d.

Find a logical expreSSion in a’ b. C. d

expreSSion in a, b. C‘ d

3. Usin

g the 16 truth values for (a,b,c,d), fill in the 16

POSSlbllities of each ‘bl . row truth table below for all the

Poss. c relation. Note. Don’t mak

the table When you‘r- l . – a me read a mess; only fill in

0. sure you have it right.

See next page