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Section 2.1 Equivalence

equivalent statements
statements \(A,B\) such that \(A \lgcbicond B\) is a tautology
\(A \lgcequiv B\)
statements \(A\) and \(B\) are equivalent

Worked Example 2.1.2. Testing logical equivalence.

Demonstrate that the following are equivalent statements.
\(A\text{:}\) If it’s nice outside, I will ride my bike.
\(B\text{:}\) It’s not nice outside, or I will ride my bike.
Solution.
Let \(p\) represent the substatement “it’s nice outside,” and let \(q\) represent the substatement “I will ride my bike.” Then the equivalence we want to establish is
\begin{equation*} p \lgccond q \lgcequiv \lgcnot p \lgcor q\text{.} \end{equation*}
We can analyze the truth tables of both statements in the same table.
\(p\) \(q\) \(\lgcnot p\) \(\lgcnot p \lgcor q\) \(p \lgccond q\)
\(\lgctrue\) \(\lgctrue\) \(\lgcfalse\) \(\lgctrue\) \(\lgctrue\)
\(\lgctrue\) \(\lgcfalse\) \(\lgcfalse\) \(\lgcfalse\) \(\lgcfalse\)
\(\lgcfalse\) \(\lgctrue\) \(\lgctrue\) \(\lgctrue\) \(\lgctrue\)
\(\lgcfalse\) \(\lgcfalse\) \(\lgctrue\) \(\lgctrue\) \(\lgctrue\)
We see that the two statements always have the same truth value in all rows of the truth table, so they are equivalent.

Note 2.1.3.

Worked Example 2.1.2 shows that the basic conditional connective “if … then …” can be constructed out of the basic connectives “not” and “or”.

Worked Example 2.1.4.

Demonstrate the equivalence \(p \lgcbicond q \lgcequiv \lgcnot p \lgcbicond \lgcnot q\text{.}\)
Solution.
Again we build a truth table, and see that the “output” columns for the two statements are identical.
\(p\) \(q\) \(\lgcnot p\) \(\lgcnot q\) \(p \lgcbicond q\) \(\lgcnot p \lgcbicond \lgcnot q\)
\(\lgctrue\) \(\lgctrue\) \(\lgcfalse\) \(\lgcfalse\) \(\lgctrue\) \(\lgctrue\)
\(\lgctrue\) \(\lgcfalse\) \(\lgcfalse\) \(\lgctrue\) \(\lgcfalse\) \(\lgcfalse\)
\(\lgcfalse\) \(\lgctrue\) \(\lgctrue\) \(\lgcfalse\) \(\lgcfalse\) \(\lgcfalse\)
\(\lgcfalse\) \(\lgcfalse\) \(\lgctrue\) \(\lgctrue\) \(\lgctrue\) \(\lgctrue\)

Check your understanding.

Thinking in terms of truth tables, consider why each of the statements of Proposition 2.1.5 holds.