Equivalence

Definition

Equivalence means two propositions have the same truth value.

Basic form:

P if and only if Q

This means:

P is true exactly when Q is true
P is false exactly when Q is false

Equivalence says two propositions match in truth value.

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Simple Example

Example:

A number is even if and only if it is divisible by 2.

Let:

P = number is even
Q = number is divisible by 2

Then:

P if and only if Q

The two statements are true together or false together.

Equivalent statements stand or fall together.

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Truth Pattern

Equivalence has this truth pattern:

P      Q      P iff Q
true   true   true
true   false  false
false  true   false
false  false  true

It is true when both sides match.

It is false when they differ.

Equivalence checks whether truth values are the same.

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Equivalence As Two Implications

Equivalence can be understood as two implications.

P iff Q

means:

If P, then Q
If Q, then P

Both directions must hold.

If only one direction holds, the statements are not equivalent.

Equivalence requires implication in both directions.

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Equivalence vs Equality

Equivalence is not always the same as equality.

Equality usually says two values or objects are the same.

Example:

x = 5

Equivalence says two propositions have the same truth behavior.

Example:

P iff Q

Equality compares values; equivalence compares truth behavior.

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Final Summary

• Equivalence means two propositions have the same truth value.
• It is often read as "if and only if."
• It is true when both propositions match.
• It is false when their truth values differ.
• Equivalence can be understood as implication in both directions.
• Equivalence is different from equality.