15 The tautologies in the next exercise will be especially useful when proving t

Question

15 The tautologies in the next exercise will be especially useful when proving the theorems concerning scts in Chapter 3. ASSIGNMENT 6. A. Show that the following statements are tautologies. LI. (png) P (and (p^q) q). 10, (-r (w^^ w)) mis is the basis for the indirect proof.) r. Now use the above tautologies (in the given order) to answer the following: 1. If you know "John is here and Judy is here," then you know"on haxo 2. If you know "John is here and Judy is here," then you know" 3. If you know "John is here," then you know . 4. "Billy has gone home or Terry is still here, or Joe has not yet arived." is logically equivalent to B. the statement "Billy has gone home, and Terry is still here and Joe has not yet arrived." is logically equivalent to the statement"Bana 5. et amived "Susie is 5 or Josie is 9, and Susie is 5 or Maggie is 7", is logically equivalent to the statement 6. 7. "Bill is 5 ft.7 in tall, and Mary is 4 f. 10 in. tall or Gordon is 4 ft. 10 in. tall," is logically equivalent to the statement" 8. "If either two integers are both odd or they are both even, then their sum is even" is logially equivalent to the statement " "Either Timmy is not 5 or Peggy is not 9, and Jimmy is not 2 or Maggie is not 7," is logically equivalent to the statement " 9.

Explanation / Answer

8A : We will use p->q is same as !p v q

(p v q) -> r <-> (p->r) ^ (q->r)

(p v q) -> r

<-> !(p v q ) v r

<-> (!p ^ !q ) v r (using Demorgan's law)

<-> !p v r ^ !q v r (distributive law)

<-> p-> r ^ q -> r

9A

(p v q) ^ (r v s)

<-> (p ^ (r v s )) v (q ^ (r v s)) (using distributive law)

<-> ((p^r) v (p^ s)) v ((q ^ r) v (q^s)) (using distributive law)

10A

!r ->(w ^ !w)

-> !(!r) v (w ^ !w) (a->b is same as !a v b )

-> r v F (a^!a is also False)

-> r (a v False is a )

8B

if sum of both integers is even then their sum is even and if the sum of both integers is odd , their sum is even.

9B

Either Timmy is not 5 and Jimmy is not 2 or Timmy is not 5 and Maggie is not 7 or Peggy is not 9 and Jimmy is not 2 or Peggy is not 9 and Maggie is not 7

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