Complete Guide to Biconditionals: Definitions and Usage

A biconditional joins two statements with if and only if, asserting each implies the other. It is true only when both parts share the same truth value, which is why a good definition is always a biconditional.

• ( p rightarrow q ): If p, then q.
• ( q rightarrow p ): If q, then p. For the biconditional to be true, both these conditions should hold.

Worked examples: biconditional statements

Practice questions on biconditionals

1. For the definition “A line is perpendicular to another if and only if they form a (90^circ) angle,” write the forward and backward readings.
2. Explain why the statement “A figure is a rectangle if it has four right angles” is not a biconditional.

1. Forward reading: If a line is perpendicular to another, they form a (90^circ) angle. Backward reading: If two lines form a (90^circ) angle, one line is perpendicular to the other.
2. The statement is only one way. It states a condition for a figure to be a rectangle but doesn’t clarify if having four right angles is the only criterion or if there are others, nor does it state the reverse (that if a figure has characteristics other than four right angles, it can’t be a rectangle). Thus, it’s not a biconditional.

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