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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.

Table of Contents

Corollaries: 1

1. Proposition: Associativity of Conjunction

Mentioned in:

Branches: 1
Corollaries: 2 3
Definitions: 4 5 6 7 8 9 10
Examples: 11
Lemmas: 12 13 14 15 16
Parts: 17
Proofs: 18 19 20 21 22 23 24 25 26 27 28 29 30
Propositions: 31

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References

Bibliography

1. Kohar, Richard: "Basic Discrete Mathematics, Logic, Set Theory & Probability", World Scientific, 2016