NOR — Definition, Formula & Examples

NOR is a logical connective that combines two statements and produces true only when both statements are false. It is equivalent to negating the result of an OR operation.

For propositions $p$ and $q$ , the NOR operation, written $p \downarrow q$ , is defined as $\lnot(p \lor q)$ . It evaluates to true if and only if both $p$ and $q$ are false.

Key Formula

p \downarrow q \;\equiv\; \lnot(p \lor q)

Where:

$p$ = First proposition
$q$ = Second proposition
$\downarrow$ = The NOR (Peirce arrow) operator
$\lor$ = Logical OR (disjunction)
$\lnot$ = Logical NOT (negation)

How It Works

To evaluate a NOR expression, first determine the truth values of both component statements. Then apply OR to those values, and finally negate the result. If either or both statements are true, the NOR is false. The only way NOR returns true is when both inputs are false. NOR is called a functionally complete connective because every other logical operation (AND, OR, NOT, etc.) can be built using NOR gates alone, which makes it especially important in digital circuit design.

Worked Example

Problem: Let p = "It is raining" (false) and q = "It is snowing" (false). Evaluate p NOR q.

Step 1: Compute the disjunction (OR) of p and q.

p \lor q = F \lor F = F

Step 2: Negate the result to get the NOR value.

p \downarrow q = \lnot(F) = T

Answer: p NOR q is true. Since neither "it is raining" nor "it is snowing" is true, the NOR of the two statements is true.

Why It Matters

NOR gates are one of the two universal logic gates (along with NAND) used to build entire computer processors. In discrete mathematics courses, understanding NOR helps you see how a single connective can replace all others, deepening your grasp of logical equivalence and proof techniques.

Common Mistakes

Mistake: Confusing NOR with OR — thinking NOR is true when at least one input is true.

Correction: NOR is the exact opposite of OR. It is true only when both inputs are false. Always remember: NOR = NOT OR.