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Direct proof

In a direct proof of P(x) → Q(x) for all x ∈ S, we consider an arbitrary element x ∈ S for which P(x) is true and show that Q(x) is true for this element x.

premises

the statements that imply the conclusion

Conclusion

Tells if your hypothesis was correct; supported or not supported

Theorem

A statement that is proved by reasoning deductively from already accepted statements.

A ∧ B

Prove A, then prove B separately. You have to prove both.

A ∨ B

Either prove A, prove B, or assume ¬A to prove B. This works because A ∨ B is equivalent to ¬A ⇒ B.

¬A

Assume A and get a contradiction.

A ⇒ B

Assume A and try to prove B.

A ⇔ B

Prove A ⇒ B and then prove B ⇒ A separately, or use a chain of "if and only if" steps to turn your goal into something true.

∀ x∈S. P(x)

Let an arbitrary x∈S be given, and prove P(x).

∃ x∈S. P(x)

Come up with x∈S so that P(x) is true.

Syllogism

type of deductive argument in which the conclusion connects on category with another, with one conclusion, two premises, and three terms

Premises [of the argument]

The part of the syllogism before the conclusion; the conditional statements that are used in syllogism