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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.

| | | | | created: 2014-06-22 19:24:58 | modified: 2018-05-04 23:35:45 | by: bookofproofs | references: [7838]