Logical conjunction

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This page belongs to resource collections on Logic and Inquiry.

Logical conjunction is an operation on two logical values, typically the values of two propositions, that produces a value of true if and only if both of its operands are true.

A logical conjunction of propositions p and q may be written in various ways.  Among the most common are these:

  • p ~\mathrm{and}~ q
  • p \land q
  • p \cdot q
  • p~q
  • pq

A truth table for p \land q appears below:

\text{Logical Conjunction}
p q p \land q
\mathrm{F} \mathrm{F} \mathrm{F}
\mathrm{F} \mathrm{T} \mathrm{F}
\mathrm{T} \mathrm{F} \mathrm{F}
\mathrm{T} \mathrm{T} \mathrm{T}


A logical graph for p \land q is drawn as two letters attached to a root node:

Cactus Graph Existential P And Q.jpg

Written as a string, this is just the concatenation pq.  The proposition pq may be taken as a Boolean function f(p, q) having the abstract type f : \mathbb{B} \times \mathbb{B} \to \mathbb{B}, where \mathbb{B} = \{ 0, 1 \} is interpreted in such a way that 0 means \mathrm{false} and 1 means \mathrm{true}.

A Venn diagram for p \land q indicates the region, in this case a single cell, where pq is true by means of a distinct color or shading, as shown below:

Venn Diagram P And Q.jpg

Syllabus

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Logical operators

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Portions of the above article were adapted from the following sources under the GNU Free Documentation License, under other applicable licenses, or by permission of the copyright holders.