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Subsection 1.2 Equivalent Statements

Given a statement \(p\text{,}\) we say that a statement \(q\) is equivalent to \(p\) if they have the same truth value. We denote equivalent statements, \(p\) and \(q\text{,}\) by

\begin{equation*} p \equiv q\text{.} \end{equation*}

For example, \(\neg(p\vee q)\equiv \neg p\wedge \neg q\) because for corresponding values of \(p\) and \(q\text{,}\) these compound statements have the same truth results. This can be seen by examining the truth tables below:

Because equivalent statements have the same truth value, we can interchange one for the other without losing the meaning of a statement. Let's try the following example. You may find the following website helpful for entering in the symbols used in these calculators Propositional Calculus 1 .

Determine which of the following statements is equivalent to \(\neg(p\wedge q)\text{:}\)

  1. \(\displaystyle \neg(p\vee q)\)

  2. \(\displaystyle \neg p\vee \neg q\)

  3. \(\displaystyle \neg p\wedge \neg q\)

Solution.

  1. Not equivalent

  2. Equivalent

  3. not equivalent

The examples above are called DeMorgan’s Laws. Summarizing DeMorgan’s Laws:

  1. \(\displaystyle \neg(p\vee q)\equiv \neg p\wedge \neg q\)

  2. \(\displaystyle \neg(p\wedge q)\equiv \neg p\vee \neg q\)

There are many other equivalent statements, as we will see in the next section.

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