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Level 166

Direct Proof & Proof by Contrapositive


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Axiom
A true mathematical statement whose truth is accepted without proof.
Theorem
A true mathematical statement whose truth can be verified. To be considered a theorem as opposed to a result, the statement should be particularly significant or interesting.
Result
Like a theorem, a result is a mathematical statement whose truth can be verified. Result is used when the statement is not especially significant or interesting.
Corollary
A mathematical result that can be deduced from, and is thereby a consequence of, some earlier result.
Lemma
A mathematical result that is useful in establishing the truth of some other result; a "helping result."
Trivial Proof
For a P → Q statement, when Q is always true regardless of P, the proof is trivial. This is almost never encountered in mathematics.
Vacuous Proof
For a P → Q statement, when P is always false, the statement is necessarily true and the proof is vacuous. This is encountered in mathematics as it is an easy mistake to make.
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.
Even
An integer n is defined to be even if n=2k for some integer k.
Odd
An integer n is defined to be odd if n=2k+1 for some integer k.
Contrapositive
The contrapositive of the implication P → Q is the implication (~Q) → (~P).
Proof by Contrapositive
A direct proof of an implication's contrapositive. This uses the fact that for every two statements P and Q, the implication P → Q and its contrapositive are logically equivalent.
Proof by Cases
When an element possesses one of two or more properties, it can be useful to divide the proof into parts called cases, i.e.
Subcases
Cases can be divided further into subcases, i.e.
Parity
In mathematics, it is the quality (of a number) of being even or odd. Since parity refers to the possession one of two properties, any result which refers to parity is likely to be solved by cases.
Of the same parity
Two numbers with the same parity are both even or both odd. Any result which refers to parity is likely to be solved by cases.
Of opposite parity
For two numbers, with "opposite parity" indicates one is even and the other is odd. Any result which refers to parity is likely to be solved by cases.
Without Loss of Generality (WLOG)
Used to indicate that the proofs of two situations are similar, so the proof of only one of these is needed.
2 | (x - y)
Two integers x and y are of the same parity if and only if ________.
Syllogism
type of deductive argument in which the conclusion connects on category with another, with one conclusion, two premises, and three terms