Questions: Biconditional Statements

Question 1

Which of the following is a true biconditional statement?

• A. A number is positive if and only if it is greater than 1
• B. A number is even if and only if it is divisible by 2
• C. A shape is a rectangle if and only if it has four sides
• D. An animal is a bird if and only if it can fly

Answer: (B)

A number is even if and only if it is divisible by 2 — both directions hold: even implies divisible by 2, and divisible by 2 implies even. Option A fails because 0.5 is positive but not greater than 1. Option C fails because trapezoids have four sides but are not rectangles. Option D fails because penguins are birds but cannot fly, and bats can fly but are not birds.

Question 2

The statement 'P if and mainly if Q' is true when P is true and Q is false.

• T. True
• F. False

Answer: False

A biconditional P ↔ Q is true exactly when P and Q have the same truth value — both true or both false. When P is true and Q is false (or vice versa), the biconditional is false. This is because one of the two directions (P → Q or Q → P) fails.

Question 3

Explain why the statement 'A quadrilateral is a square if and only if it has four equal sides' is false, and correct it.

Model answer: It is false because a rhombus has four equal sides but is not a square (it may lack right angles). The converse direction fails: four equal sides does not guarantee a square. A correct version: 'A quadrilateral is a square if and only if it has four equal sides and four right angles.'

Testing a biconditional requires checking both directions. The forward direction ('if square, then four equal sides') is true. The reverse direction ('if four equal sides, then square') is false — the rhombus is a counterexample. The corrected biconditional adds the condition about right angles, which eliminates the rhombus counterexample.