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Level 167

Direct Proof


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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