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All the binary connectives we have learned so far – conjunction, disjunction and exclusive disjunction, were commutative – the order of the propositions connected by them did not matter. Now, we will learn an important binary connective which is not commutative – the implication.

Definition: Implication

Let \(L\) be a set of propositions with the interpretation $I$ and the corresponding valuation function $[[]]_I$. An implication “\(\Rightarrow\)” is a Boolean function

\[\Rightarrow :=\begin{cases}L \times L & \mapsto L \\
(x,y) &\mapsto x \Rightarrow y.
\end{cases}\]

defined by the following truth table:

Truth Table of the Implication
$[[x]]_I$ $[[y]]_I$ $[[x \Rightarrow y]]_I$
\(1\) \(1\) \(1\)
\(0\) \(1\) \(1\)
\(1\) \(0\) \(0\)
\(0\) \(0\) \(1\)

We read the implication $x\Rightarrow y$

if $x$ then $y$”.

Notes

| | | | | created: 2015-06-06 20:22:00 | modified: 2018-05-14 22:24:12 | by: bookofproofs | references: [7838]

1.Lemma: Implication as a Disjunction

2.Lemma: Negation of an Implication

3.Definition: Contrapositive

Edit or AddNotationAxiomatic Method

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Bibliography (further reading)

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