Section 5.1 Some Common Situations to Use Cases
When using cases in a proof, the main rule is that the cases must be chosen so that they exhaust all possibilities for an object \(x\) in the hypothesis of the original proposition. Following are some common uses of cases in proofs.
- When the hypothesis is, “\(\boldsymbol{n}\) is an integer.”
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Case 1: \(n\) is an even integer.
Case 2: \(n\) is an odd integer.
- When the hypothesis is, “\(\boldsymbol{m}\) and \(\boldsymbol{n}\) are integers.”
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Case 1: \(m\) and \(n\) are even.
Case 2: \(m\) is even and \(n\) is odd.
Case 3: \(m\) is odd and \(n\) is even.
Case 4: \(m\) and \(n\) are both odd.
- When the hypothesis is, “\(x\) is a real number.”
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Case 1: \(x\) is irrational.
Case 2: \(x\) is irrational.
- When the hypothesis is, “\(\boldsymbol{x}\) is a real number.”
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Case 1: \(a = b\text{.}\)
Case 2: \(a \ne b\text{.}\)
OR
Case 1: \(a \gt b\text{.}\)
Case 2: \(a = b\text{.}\)
Case 3: \(a \lt b\text{.}\)
- When the hypothesis is, “\(a\) and \(b\) are real numbers.”
-
Case 1: \(a = b\text{.}\)
Case 2: \(a \ne b\text{.}\)
OR
Case 1: \(a \gt b\text{.}\)
Case 2: \(a = b\text{.}\)
Case 3: \(a \lt b\text{.}\)