**Definition**: Conjunction

Let \(L\) be a set of propositions with the interpretation $I$ and the corresponding valuation function $[[]]_I$. A **conjunction** “\(\wedge \)” is a Boolean function

\[\wedge :=\begin{cases}L \times L & \mapsto L \\

(x,y) &\mapsto x \wedge y.

\end{cases}\]

defined by the following truth table:

$[[x]]_I$ | $[[y]]_I$ | $[[x \wedge y]]_I$ |
---|---|---|

\(1\) | \(1\) | \(1\) |

\(0\) | \(1\) | \(0\) |

\(1\) | \(0\) | \(0\) |

\(0\) | \(0\) | \(0\) |

We read the conjunction $x\wedge y$

“$x$ *and* $y$”.

### Notes

- The conjunction of two propositions is only true if they are both true, otherwise it is false.
- The standard English conjunction is
*and*. But there are some others, which are the same logically:*but*,*however*,*although*,*though*,*even though*,*moreover*,*furthermore*, and*whereas*, e.g.- “I passed logic,
*and*I did not pass calculus.” - “I passed logic,
*but*I did not pass calculus.” - “I passed logic,
*however*I did not pass calculus.” - “I passed logic,
*although*I did not pass calculus.” - “I passed logic,
*though*I did not pass calculus.” - etc.

- “I passed logic,

| | | | | Contributors: *bookofproofs* | References: [7838]

## 1.**Corollary**: Commutativity of Conjunction

## 2.**Proposition**: Associativity of Conjunction

(none)

[7838] **Kohar, Richard**: “Basic Discrete Mathematics, Logic, Set Theory & Probability”, World Scientific, 2016

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