EF Direct proof

Section 6.3 Direct proof

The argument

A \to C_{1}, C_{1} \to C_{2}, \dots, C_{m - 1} \to C_{m}, C_{m} \to B ∴ A \to B

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To prove $P \Rightarrow Q,$ start by assuming that $P$ is true. Then, through a sequence of (appropriately justified) intermediate conclusions, arrive at $Q$ as a final conclusion.
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To prove $(\forall x) (P (x) \Rightarrow Q (x)),$ start by assuming that $x$ is an arbitrary but unspecified element in the domain such that $P (x)$ is true. The first sentence in your argument should be: “Suppose $x$ is a such that $P (x)$ ”, where the blank is filled in by the definition of the domain of $x .$ Then, through a sequence of (appropriately justified) intermediate conclusions that do not depend on knowing the specific object $x$ in the domain, arrive at $Q (x)$ as a conclusion.

Prove: If

n

is even, then

n^{2}

even.

Let

P (n)

represent the predicate “

n

is even” and let

Q (n)

represent the predicate “

n^{2}

is even”, with domain the integers.

Suppose that

n

is an arbitrary (but unspecified) integer such that

n

is even. Then there exists an integer

m

such that

n = 2 m,

and so

n^{2} = 4 m^{2} = 2 (2 m)

is even.

Elementary Foundations: An introduction to topics in discrete mathematics