Question

For all real numbers x, if x² ≥ 1 then x > 0. What is the negation, converse, inverse, and contrapositive of the statement?

1) The negation is: ∃ real numbers x, if x² ≥ 1 then x ≤ 0.
2) The converse is: ∀ real numbers x, if x > 0 then x² ≥ 1.
3) The inverse is: ∀ real numbers x, if x² < 1 then x ≤ 0.
4) The contrapositive is: ∀ real numbers x, if x ≤ 0 then x² < 1.

Answer 1

The negation, converse, inverse, and contrapositive of the given statement are explained.

Step-by-step explanation:

The given statement is: For all real numbers x, if x² ≥ 1 then x > 0. Let's analyze the negation, converse, inverse, and contrapositive of this statement:

1. Negation: The negation of the statement is: There exist real numbers x, such that if x² ≥ 1 then x ≤ 0. This means that there are some real numbers for which x² is greater than or equal to 1, but x is less than or equal to 0.
2. Converse: The converse of the statement is: For all real numbers x, if x > 0 then x² ≥ 1. This means that if x is greater than 0, then x² is greater than or equal to 1.
3. Inverse: The inverse of the statement is: For all real numbers x, if x² < 1 then x ≤ 0. This means that if x² is less than 1, then x is less than or equal to 0.
4. Contrapositive: The contrapositive of the statement is: For all real numbers x, if x ≤ 0 then x² < 1. This means that if x is less than or equal to 0, then x² is less than 1.