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FUNDAMENTOS DE COMPUTACION

FUNDAMENTOS DE COMPUTACION

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Choose the main technique to prove the following statement:

For every integer m, m(m+1)(m+2) is an integer multiple of 3.

T5: Proof by cases to show $\displaystyle \left(\left(\bigvee_{i=1}^n p_i \right) \to q \right) \equiv T$ \displaystyle \left(\left(\bigvee_{i=1}^n p_i \right) \to q \right) \equiv T

T6: Constructive proof to show $\exists x P(x) \equiv T$ \exists x P(x) \equiv T

T7: Two-step proof to show $\exists ! x P(x) \equiv T$ \exists ! x P(x) \equiv T

T1: Direct proof to show $(p \to q) \equiv T$ (p \to q) \equiv T

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Choose the main technique to prove the following statement:

The sum of two rational numbers is rational.

T5: Proof by cases to show $\displaystyle \left(\left(\bigvee_{i=1}^n p_i \right) \to q \right) \equiv T$ \displaystyle \left(\left(\bigvee_{i=1}^n p_i \right) \to q \right) \equiv T

T6: Constructive proof to show $\exists x P(x) \equiv T$ \exists x P(x) \equiv T

T7: Two-step proof to show $\exists ! x P(x) \equiv T$ \exists ! x P(x) \equiv T

T1: Direct proof to show $(p \to q) \equiv T$ (p \to q) \equiv T

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Choose the main technique to prove the following statement:

If n is an integer and $2 \leq n \leq 4$ 2 \leq n \leq 4, then $n^2 \geq 2^n$ n^2 \geq 2^n.

T5: Proof by cases to show $\displaystyle \left(\left(\bigvee_{i=1}^n p_i \right) \to q \right) \equiv T$ \displaystyle \left(\left(\bigvee_{i=1}^n p_i \right) \to q \right) \equiv T

T6: Constructive proof to show $\exists x P(x) \equiv T$ \exists x P(x) \equiv T

T7: Two-step proof to show $\exists ! x P(x) \equiv T$ \exists ! x P(x) \equiv T

T1: Direct proof to show $(p \to q) \equiv T$ (p \to q) \equiv T

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To prove: If n^2 is odd, then n is odd.

In the following statements, first, determine whether a statement shall be used in the proof, then, if the statement shall be used, determine the right sequence to present it.

Assume that n is even.

Assume that n is odd.

n^2 is even

n = 2k+1 for an integer k.

n^2 = (2k+1)^2 = 2(2k^2+2k) + 1.

proof

n = 2k for an integer k

Consider an arbitrary integer n

n^2 is odd.

n^2 = (2k)^2 = 2(2k).

QED

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To prove: If n is odd, then n^2 is odd.

In the following statements, first, determine whether a statement shall be used in the proof, then, if the statement shall be used, determine the right sequence to present it.

Assume that n is even.

Assume that n is odd.

n^2 is even

n = 2k+1 for an integer k.

n^2 = (2k+1)^2 = 2(2k^2+2k) + 1.

proof

n = 2k for an integer k

Consider an arbitrary integer n

n^2 is odd.

n^2 = (2k)^2 = 2(2k).

QED

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Choose the main technique to prove the following statement.

If 5n^2+6n+7 is odd, then n is even.

T3: indirect proof by contradiction to show $p \equiv T$ p \equiv T

T2: indirect proof by contraposition to show $p \to q \equiv T$ p \to q \equiv T

T1: direct proof to show $p \to q \equiv T$ p \to q \equiv T

T4: two-step proof to show $p \leftrightarrow q \equiv T$ p \leftrightarrow q \equiv T

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Choose the main technique to prove the following statement.

If n is even, then n^2 is even.

T3: indirect proof by contradiction to show $p \equiv T$ p \equiv T

T1: direct proof to show $p \to q \equiv T$ p \to q \equiv T

T2: indirect proof by contrapostion to show $p \to q \equiv T$ p \to q \equiv T

T4: two-step proof to show $p \leftrightarrow q \equiv T$ p \leftrightarrow q \equiv T

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Choose the main technique to prove the following statement.

n is even if and only if 5n^2+6n+7 is odd.

T3: indirect proof by contradiction to show $p \equiv T$ p \equiv T

T1: direct proof to show $p \to q \equiv T$ p \to q \equiv T

T2: indirect proof by contrapostion to show $p \to q \equiv T$ p \to q \equiv T

T4: two-step proof to show $p \leftrightarrow q \equiv T$ p \leftrightarrow q \equiv T

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Choose the main technique to prove the following statement.

$\sqrt{3}$ \sqrt{3} is irrational.

T3: indirect proof by contradiction to show $p \equiv T$ p \equiv T

100%

T1: direct proof to show $p \to q \equiv T$ p \to q \equiv T

T2: indirect proof by contrapostion to show $p \to q \equiv T$ p \to q \equiv T

T4: two-step proof to show $p \leftrightarrow q \equiv T$ p \leftrightarrow q \equiv T

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Choose the main technique to prove the following statement.

If n^2 is even, then n is even.

T3: indirect proof by contradiction to show $p \equiv T$ p \equiv T

T1: direct proof to show $p \to q \equiv T$ p \to q \equiv T

T2: indirect proof by contrapostion to show $p \to q \equiv T$ p \to q \equiv T

T4: two-step proof to show $p \leftrightarrow q \equiv T$ p \leftrightarrow q \equiv T

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