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
$[[x]]_I$ 
$[[y]]_I$ 
$[[x \Rightarrow y]]_I$ 
\(1\) 
\(1\) 
\(1\) 
\(0\) 
\(1\) 
\(1\) 
\(1\) 
\(0\) 
\(0\) 

\(0\) 
\(0\) 
We read the implication $x\Rightarrow y$
“if $x$ then $y$”.
Notes
 It is apparent from the truth table that the implication is not commutative  the order of the propositions connected by it counts. Therefore, both propositions have specific names  the first one is called the antecedent, the second one the consequent.
 The implication of two propositions is only false if a true antecedent implies a false consequent. This is called a contradiction. Note that a false antecedent can imply both a true and a false consequent, without creating a contradiction.
 But if a true antecedent implies a true consequent, we say it is a valid argument.
 There are more different readings of the implication $x\Rightarrow y$:
 If $x$, then $y$.
 $y$ follows from $x$.
 $x$ is sufficient for $y$.
 $y$ is necessary for $x$.
 $y$, if $x$.
 $x$, only if $y$.
 It is helpful to think about the implication as a cause and effect chain.
Table of Contents
 Lemma: Implication as a Disjunction
 Lemma: Negation of an Implication
 Definition: Contrapositive
Mentioned in:
Branches: 1
Definitions: 2 3
Examples: 4
Explanations: 5
Lemmas: 6 7 8 9 10 11 12 13 14
Parts: 15
Proofs: 16 17 18 19 20 21 22 23 24 25 26
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References
Bibliography
 Kohar, Richard: "Basic Discrete Mathematics, Logic, Set Theory & Probability", World Scientific, 2016