Determine the truth value of each of these statements if the domain of each variable consists of all real numbers. [4 marks] a)∃x(x2 =2) b)∃x(x2 =−1) c) ∀x (x2 + 2 ≥ 1) d) ∀x (x2 =x)

a) ∃x(x² = 2) - This statement is true. There exists a real number x such that x² = 2. Specifically, x can be √2 or -√2.

b) ∃x(x² = -1) - This statement is false. There is no real number x such that x² = -1. The square of any real number is always non-negative.

c) ∀x (x² + 2 ≥ 1) - This statement is true. For all real numbers x, x² + 2 is always greater than or equal to 1. This is because x² is always non-negative for real numbers, so x² + 2 is always at least 2.

d) ∀x (x² = x) - This statement is false. For all real numbers x, x² is not always equal to x. This is only true for x = 0 or x = 1.

a) The statement ∃x(x² = 2) is true. This is because there exists a real number x (specifically, x = sqrt(2) or x = -sqrt(2)) such that x² = 2.

b) The statement ∃x(x² = -1) is false. There is no real number x such that x² = -1. This is because the square of any real number is always non-negative.

c) The statement ∀x (x² + 2 ≥ 1) is true. This is because for all real numbers x, x² is always non-negative. Therefore, x² + 2 is always greater than or equal to 1.

d) The statement ∀x (x² = x) is false. This is because for all real numbers x, x² = x only when x = 0 or x = 1. For any other real number, x² ≠ x.