Question

Determine the truth value of each of these statements if the domain for all variables consists of all integers.

a) ∀n(n²≥0)
b) ∃n(n²=2)
c) ∀n(n²≥n)
d) ∃n(n²<0)

Answer 1

a) The statement is true since any integer squared is non-negative. b) The statement is false since there is no integer solution for n squared equals 2. c) The statement is true since any integer squared is greater than or equal to the integer. d) The statement is false since there is no integer solution for n squared less than 0.

Step-by-step explanation:

a) The statement ∀n(n²≥0) is true. This statement asserts that for every integer value of n, n squared is greater than or equal to 0. Since any integer squared will always be non-negative, the statement is true.

b) The statement ∃n(n²=2) is false. This statement asserts that there exists an integer value of n such that n squared equals 2. However, since the square of any integer will always be a non-negative value, there is no integer solution for n in this case.

c) The statement ∀n(n²≥n) is true. This statement asserts that for every integer value of n, n squared is greater than or equal to n. Since n squared will always be greater than or equal to n for any integer value of n, the statement is true.

d) The statement ∃n(n²<0) is false. This statement asserts that there exists an integer value of n such that n squared is less than 0. However, since the square of any integer will always be a non-negative value, there is no integer solution for n in this case.