Let Q(x) be the statement \"x + l lessthanorequalto 2x\". If the domain consists

Question

Let Q(x) be the statement "x + l lessthanorequalto 2x". If the domain consists of all integers, what are the truth values of the following statements? Explain your answers (by examples). (a) Q(l) Solution: instead of x plug in '1': we'll get "1 + 1 lessthanorequalto 2*1" which is valid: so Q(l) is true. (b) Q(-l) (c) forall x Q(x) (d) exists x Q(x) exists x (x) Prove the following tautologies by starting with the left side and finding a series of equivalent wffs that will convert the left side into the right side. You may use any of the equivalencies from the handout (also found on iCollege folder in extras). Note that you are not allowed to build a truth table for this particular exercise. a. (A logicaland B') logicaland C doubleheadarrow (A logicaland C) logicaland B' b. b. (A logicalor B) logicaland (A logicalor B') doubleheadarrow A Solution: a. (A logicaland B') logicaland C b. b. (A logicalor B) logicaland (A logicalor B')

Explanation / Answer

(According to Chegg policy only four subquestions will be answered. Please post the remaining in another question)

Q(x): x+1 <= 2x

(a) Q(1)

Substituting x = 1 => x+1 = 2 <=2*1 = 2

Thus Q(1) is true.

(b) Q(-1)

Substituting x = -1 => x+1 = 0 > 2*-1 = -2

Thus Q(-1) is false.

(c) x, Q(x)

This translates to "for all integers, Q is true"

However we already saw Q(-1) being false

So this statement is false.

(d) x, Q(x)

This translates to "for some integer, Q is true"

We already saw Q(1) being true

So this statement is true.

(e) x, ~Q(x)

This translates to "for some integer, Q is not true"

We already saw Q(-1) being not true

So this statement is true.

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