In this section, we learn to write proofs for existential statements and universal statements. We also learn how to write counterexamples.
It is very important to remember that the negation of a universal statement is an existential statement. This means that to disprove a universal statement, we only need to find ONE counterexample!
Example:
Proposition: For every positive integer n, n^2 is odd.
Counterexample: Note that 2 is a positive integer, but 2^2 = 4 is even. Therefore, the proposition is false.
You DO NOT want to use a generic variable to show that this proposition is false, because it logically opposes the negation of a universal statement being an existential one!
The notes for this section are available here.
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