So far in this course, there seems to be two categories of proofs in terms of structure. One where we prove that something holds for all cases. Another where we prove that there is at least one such case that holds. When we attempt to disprove a statement, we will negate it and prove that the negation is true. It's negation will generally consist of existential and/or universal quantifies also, simply with existential switched to universal and vice-versa. Therefore, negations do not really have a different proof structure than existential or universal.
When proving a statement that includes an existential quantifier, we need to pick a value that satisfy the required set. If that value is currently unknown and we may be able to derive it after going through the proof a bit, we would leave a blank line to fill it in later. Then the rest of the proof for the existential portion will be to prove the statement inside the world where we picked the value. If the statement holds for that value then we can conclude that there exist a value which satisfy the statement and thus conclude our proof.
When proving a statement that includes a universal quantifier, we need to assume a variable is a typical member of the set. Then in the world, we need to assume that the antecedent is true. Using the antecedent, we need to derive the consequence, and if we can, then we conclude that the antecedent implies the consequence. Since we did it for a typical member of the set, we then extrapolate and conclude that for all members of the set, the statement holds and thus conclude our proof.
Simply looking at the two, I seem to see that proofs including universal quantifiers requires a lot more assumption and bigger assumptions too than that of ones with only existential. Sometimes it seems that we need to take a big leap and conclude that it will work for any members of the set.
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