The contrapositive of the statement "If $x^2 - 1 = 0$ , then $x = -1$ or $x = + 1$ " is

If $x \neq - 1$ and $x \neq + 1$ , then $x^2 - 1 \neq 0$

The contrapositive of the statement "If $x^2 - 1 = 0$ , then $x = -1$ or $x = + 1$ " is}

If $x \neq - 1$ and $x \neq + 1$ , then $x^2 - 1 \neq 0$

If $x^2 - 1 \neq 0$ , then $x \neq - 1$ or $x \neq + 1$

If $x^2 - 1 \neq 0$ , then $x \neq - 1$ and $x \neq + 1$

If $x \neq - 1$ or $x \neq + 1$ , then $x^2 - 1 \neq 0$

If $x \neq - 1$ and $x \neq + 1$ , then $x^2 - 1 \neq 0$

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A statement is a sentence that is either true or false, but not both true and false simultaneously.

If a statement cannot be further broken down into various statements, or in simpler words if it is concrete by itself, it is called a Simple Statement.
Examples include:
- A kite is not a rhombus.
- 15 is an odd number.

If a statement can further be broken down into simpler statements so that from a main statement, we can yield more than one statement, then it is called a Compound Statement.
Consider the statement “10 is non-negative and a multiple of 5” which can be broken down into the statements: “10 is non-negative” and “10 is a multiple of 5”.

If we encounter an if-then statement i.e. ‘if a then b’, then by proving that a is true, b can be proved to be true or if we prove that b is false, then a is also false.
If we encounter a statement which says ‘a if and only if b’, then we can give reason for such a statement by showing that if a is true, then b is also true and if b is true, then a is also true.
Example:
- a: 8 is multiple of 64
- b: 8 is a factor of 64

Since one of the given statements i.e. a is true, therefore, a or b is true.