In mathematics, the symbol ↔ typically means "if and only if" or "is equivalent to".
This symbol signifies a biconditional statement, indicating that two statements are logically equivalent. In other words, each statement implies the other.
Detailed Explanation
The "if and only if" connective (often abbreviated as "iff") asserts that two statements have the same truth value. If one statement is true, the other must be true. If one statement is false, the other must be false. There's no other possibility.
Let's consider two statements, p and q. The statement p ↔ q is true only when both p and q are true or both p and q are false.
Formally, p ↔ q is equivalent to ( p → q ) ∧ ( q → p ), meaning that p implies q and q implies p.
Truth Table
Here's a truth table illustrating the behavior of ↔:
| p | q | p ↔ q |
|---|---|---|
| True | True | True |
| True | False | False |
| False | True | False |
| False | False | True |
Examples
- Example 1: A triangle has three sides if and only if it is a polygon. (The definition of a triangle dictates this equivalence.)
- Example 2: x = y if and only if x - y = 0.
- Example 3: A number is divisible by 2 if and only if it is even.
Other Names and Representations
The ↔ symbol is also sometimes referred to as:
- Equivalence
- Biconditional
- NXOR (exclusive nor)
In some contexts, the symbol ⟺ is also used to represent "if and only if".
In summary, ↔ signifies a strong connection between two statements, asserting that they are logically interchangeable. One statement being true guarantees the other is true, and one statement being false guarantees the other is false.
