Last week, we learned how to prove a statement false. As it turns out, the simplest way to prove something false is to prove the negation true. Since if the negation is true then the negation of the negation which would be the original statement, has to be false.
If a statement begins with a for all quantifier, we can quickly assess to see if it can be potentially a false statement by looking for a counter example. If the statement is an implication then we would try to think up something that satisfies the antecedent but not the consequence. If there indeed does exist one such case then we can assume the statement is false as the for all quantification is not longer met. Following this, we would negate the statement and proceed to prove the negated statement as we would for any other proofs. In the case of our example with a false for all quantified statement, we would now go forward by proving there a there exists quantification and most likely pick the example as the same one we used to figure out that the original statement was false in the first place.
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