Answer to Question #282780 in Math for Lejanie

Question

1.Let a and b be real numbers and let p, q, and r denote the following sentences:

p = (a – x < b – x); q = (a < b); r = (a/x<b/x). Indicate whether each of the following compound sentences yields a true or false statement (A) when x = 2; (B) when x = -2.

a.p ↔ q

b. p ↔ r

c. r → p

d. p →r

e.q ↔ r

f. q → p

g. r → q

h. q →r

What can you say about each of the above expressions a – h when x = 0?

2. Which of the above sentences in No. 1 become implications and which of them become equivalences, if the admissible values for x are:

a. all positive real numbers?

b.All negative real numbers?

c.All real numbers?

3.Prove the following equivalences.

a.(p → q) <=> (~p V q)

b. ~(p ↔ q) <=> (p V q)

c.(p → q) <=> (~q → ~p)

d. [(q Λ r) V p] <=> [(p V q) Λ (p V r)]