Definition:Biconditional/Semantics of Biconditional/Necessary and Sufficient

Definition

Let:

$p \iff q$

where $\iff$ denotes the biconditional operator.

Then it can be said that $p$ is necessary and sufficient for $q$.

This is a consequence of the definitions of necessary and sufficient conditions.

Examples

Arbitrary Example

Let $x$ be an integer.

For $x$ to be divisible by $6$, it is:

necessary but not sufficient for $x$ to be even

sufficient but not necessary for $x$ to be divisible by $12$

necessary and sufficient for $x$ to be both even and divisible by $3$.

Sources

• 1946: Alfred Tarski: Introduction to Logic and to the Methodology of Deductive Sciences (2nd ed.) ... (previous) ... (next): $\S \text{II}.10$: Equivalence of sentences
• 1971: Robert H. Kasriel: Undergraduate Topology ... (previous) ... (next): $\S1.2$: Some Remarks on the Use of the Connectives and, or, implies: Definition $2.1$
• 1977: Gary Chartrand: Introductory Graph Theory ... (previous) ... (next): Appendix $\text{A}.5$: Theorems and Proofs
• 1982: P.M. Cohn: Algebra Volume 1 (2nd ed.) ... (previous) ... (next): Chapter $1$: Sets and mappings: $\S 1.1$: The need for logic
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): necessary condition
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): necessary condition
• 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): condition, necessary and sufficient