Question:

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

Updated On: May 12, 2024
  • 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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The Correct Option is D

Solution and Explanation

Answer (d) If $x \neq - 1$ and $x \neq + 1$, then $x^2 - 1 \neq 0$
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Concepts Used:

Statements

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

Types of Statements:

Simple Statement

  • 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.

Compound Statement

  • 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-Then Statements

  • 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.