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Contents Index Dark Mode Prev Up Next \(\newcommand{\im}{{\mathrm{im}}}
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Subsection 1.4.6 Summary guidelines for proofs
Each sentence should set notation or be a true statement.
Each true statement should be a conclusion that can be drawn from the previous statements using a definition, computation, or result proved in class.
Do not assert the truth of statement to be proven at the beginning of a proof. Preface such statements with “We wish to prove” or something similar.
Oftentimes, a good first step is just unwinding the definitions.
To prove “if
\(p\) , then
\(q\) ” directly, start the proof by assuming
\(p\) is true. Then deduce that
\(q\) must be true. See
Proof Technique 1.4.2.1 .
To prove “if
\(p\) , then
\(q\) ” by contraposition, start the proof by assuming
\(q\) is false. Then deduce that
\(p\) must be false. See
Proof Technique 1.4.3.1 .
To prove
\(p\) by contradiction, start the proof by assuming
\(p\) is false. Then deduce a contradiction. See
Proof Technique 1.4.4.1 .
Here are some examples of what is meant by
Item 2 above.
“
\(ax\) ” is not a sentence.
“
\(ax = b\) has a solution.” is a sentence, but it is not true or false. We need to know more about
\(a\) and
\(b\text{.}\)
“Let
\(a \in \RR\) ,
\(a \neq 0\text{.}\) Then
\(ax = b\) has a solution.” is a bit better. The first sentence sets notation, but the second sentence is still neither true nor false since we have not specified the universe for
\(b\) .
“Let
\(a \in \RR\) ,
\(a \neq 0\text{.}\) Then
\(ax = b\) has a solution for every
\(b \in \RR\text{.}\) ” The first sentence sets notation. All of the notation is defined. The second sentence is true.
