I need help with these 7 discrete math proofs

2.

Let D = R. Let P(x) denote the propositional function “2x x”. Is 8x P(x) true

or false? Prove your answer is correct. If we change D to Z+, does this change the truth

value of 8x P(x)? Why or why not?

3.

Let P(x) denote the propositional function “x2 + 1 = 0”. Can we determine whether

or not 9x P(x) is true or false without further information? Justify your answer.

4.

Let D = R, and let P(x, y) denote the propositional function “x + y = 1”.

(a) Is 8x9y P(x, y) true or false? Justify your answer.

(b) Is 9y8x P(x, y) true or false? Justify your answer.

5.

Let D denote the set of all children in the world. Let P(x, y) denote the propositional

function “x plays with y.” Translate the following quoted statement into discrete math

notation, using quantifier symbols and P(x, y): “Every child plays with some child.”

6.

(a) Let

D = R. Is the quantified statement 8x9y (x > y) true or false? Prove your

answer is correct.

(b) Let D be the set of strictly negative integers. Is the quantified statement

8

x9y (x < y)

true or false? Prove your answer is correct.

7.

Let P(x, y, z) be a propositional function. Use the Generalized DeMorgan’s Laws for

Logic to negate the following:

(a) 8x9y8z P(x, y, z)

(b) 9x8y9z P(x, y, z)