Implications in a Formal Mathematical Proof

In a formal proof, implication statements are used to represent a conditional relationships between the truth of different propositions, for example

If it is raining, then the ground is wet.

In this example, we have two propositions

\begin{aligned} R &= \text{It is raining} \\ W &= \text{The ground is wet} \\ \end{aligned}

The implication statement tells us that the truth of proposition $R$ implies the truth of proposition $W$ , or that if the statement 'it is raining' is true, then the statement 'the ground is wet' is also true. We can write this more concisely as

$R \Rightarrow W$

Writing it in this way shows a implication consists of two parts: a hypothesis (or antecedent) and a conclusion (or consequent). The truth of the hypothesis implies the truth of the conclusion.

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