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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:

Truth Table of the Conjunction
$[[x]]_I$ $[[y]]_I$ $[[x \wedge y]]_I$
\(1\) \(1\) \(1\)
\(0\) \(1\) \(0\)
\(1\) \(0\) \(0\)
\(0\) \(0\) \(0\)

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.

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

1.Corollary: Commutativity of Conjunction

2.Proposition: Associativity of Conjunction


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Bibliography (further reading)

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

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