Question

A. Use truth tables to verify these equivalences

1. p∨p ≡ p

2. p∧p ≡ p

3. p∨(p∧q) ≡ p

4. p∨q ≡¬p → q

5. p∧q ≡¬(p →¬q)

6. p ↔ q ≡ (p → q)∧(q → p)

B. Determine the truth value of each of these statements. (Assume the domain of variables consist of all real numbers).

1. ∃x(x2 = 2)

2. ∃x(x + 2 = x)

3. ∀x(x2 + 2 > 0)

4. ∀x(x2 = x)

Answer 1

1.

True

2.

True

3.

True

4.

True

5.

True

6.

True

++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++========

1.
∃x: (x^2 = 2)

x^2 = 2
x = √2 = 1.4142

Since value of x exists for the condition, hence value for the above quantifier is truth.

2.

∃x: (x + 2 = x)

x + 2 = x

==> x - x = 2 not possible.

So this statement is false.

3.

∀x: (x^2 + 2 > 0)

x = 1, 2, 3...

value of x^2 will always be +ve, so this holds true,

for x = -1, -2, -3

x^2 will always be true, so true.

Hence this statement is true.

4.
∀x: (x^2 = x)

if x = 1

x^2 = 1 = x, so true.

x= 2
x^2 = 4 != x, so false.

since there is universal quantifier so, it must hold true for all values of x.

So this statement is false.

IF THERE IS ANYTHING THAT YOU DO NOT UNDERSTAND, OR NEED MORE HELP THEN PLEASE MENTION IT IN THE COMMENTS SECTION,