Hard
biconditionalif-and-only-iflogical-equivalence

What Does This 'If and Only If' Statement Imply?

Problem Statement

Statement: 'A shape is a square if and only if it has four equal sides and four right angles.' Which is true? A) All rectangles are squares. B) A shape with four equal sides must be a square. C) A square must have four equal sides and four right angles. D) Some squares don't have right angles.

Answer

C) A square must have four equal sides and four right angles

Step-by-Step Explanation

  1. 1

    'If and only if' (biconditional) means both directions are true:

  2. 2

    Direction 1: If square → four equal sides AND four right angles

  3. 3

    Direction 2: If four equal sides AND four right angles → square

  4. 4

    This creates a logical equivalence: Square ↔ (four equal sides AND four right angles)

  5. 5

    Option A: 'All rectangles are squares' — False. Rectangles have right angles but not necessarily equal sides.

  6. 6

    Option B: 'Four equal sides must be a square' — False. Also needs right angles (could be a rhombus).

  7. 7

    Option C: 'A square must have four equal sides and four right angles' — TRUE. This is Direction 1.

  8. 8

    Option D: 'Some squares don't have right angles' — False. Contradicts the definition.

  9. 9

    The biconditional means BOTH conditions are necessary and together they are sufficient.

  10. 10

    Answer: C