Complete Guide to Biconditionals: Definitions and Usage

Step-by-step Guide: Biconditionals and Definitions

Understanding Biconditionals
A biconditional is a compound statement formed by combining two conditional statements using the phrase “if and only if,” often abbreviated as “iff.” It’s denoted by the symbol \(\leftrightarrow\). A biconditional statement is true when both the conditions have the same truth value, i.e., both are true or both are false.

Structure: \( p \leftrightarrow q \) reads as “\(p\) if and only if \(q\).”

Biconditional Breakdown
When we say “\(p\) if and only if \(q\),” it encompasses two statements:

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

Biconditionals in Geometry & Definitions
In geometry, biconditionals play a pivotal role in creating clear-cut definitions. A definition in geometry often allows for both a forward and backward reading, which is precisely the nature of biconditionals.

Example: A figure is a square if and only if it has four equal sides and four right angles. Here, if a figure is a square, it must have these properties. Conversely, if a figure possesses these properties, it’s defined to be a square.

Examples

Example 1:
Consider the statement: A shape is a circle if and only if all points on the shape are equidistant from a single point called the center.

Solution:
Forward reading: If a shape is a circle, then all its points are equidistant from a center.
Backward reading: If all points on a shape are equidistant from a center, then the shape is a circle.

Example 2:
Statement: An angle is a right angle if and only if it measures \(90^\circ\).

Solution:
Forward: If an angle is a right angle, it measures \(90^\circ\).
Backward: If an angle measures \(90^\circ\), it is a right angle.

Practice Questions:

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.

Answers:

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.