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Suppose we want to prove: "If n is an even natural number, then for all x € R, x" ≥ 0." Classify each of the following as either a mistake or the beginning of a direct proof, proof by contrapos- itive, or proof by contradiction. Let n be an even natural number. Suppose there exists an x ER such that x" <0. Let n be an odd natural number, and suppose that for all x € R, x ≥ 0. Let n be an even natural number. Suppose that for all x € R, x² <0. Take an arbitrary n E N. Suppose there exists an x € R such that x" <0. Let n be any odd natural number.

Suppose we want to prove: "If n is an even natural number, then for all x € R, x" ≥ 0." Classify each of the following as either a mistake or the beginning of a direct proof, proof by contrapos- itive, or proof by contradiction. Let n be an even natural number. Suppose there exists an x ER such that x" <0. Let n be an odd natural number, and suppose that for all x € R, x ≥ 0. Let n be an even natural number. Suppose that for all x € R, x² <0. Take an arbitrary n E N. Suppose there exists an x € R such that x" <0. Let n be any odd natural number.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Suppose we want to prove: “If \( n \) is an even natural number, then for all \( x \in \mathbb{R} \), \( x^n \geq 0 \).” Classify each of the following as either a mistake or the beginning of a direct proof, proof by contrapositive, or proof by contradiction. 1. Let \( n \) be an even natural number. Suppose there exists an \( x \in \mathbb{R} \) such that \( x^n < 0 \). 2. Let \( n \) be an odd natural number, and suppose that for all \( x \in \mathbb{R} \), \( x^n \geq 0 \). 3. Let \( n \) be an even natural number. Suppose that for all \( x \in \mathbb{R} \), \( x^n < 0 \). 4. Take an arbitrary \( n \in \mathbb{N} \). Suppose there exists an \( x \in \mathbb{R} \) such that \( x^n < 0 \). 5. Let \( n \) be any odd natural number.
Transcribed Image Text:Suppose we want to prove: “If \( n \) is an even natural number, then for all \( x \in \mathbb{R} \), \( x^n \geq 0 \).” Classify each of the following as either a mistake or the beginning of a direct proof, proof by contrapositive, or proof by contradiction. 1. Let \( n \) be an even natural number. Suppose there exists an \( x \in \mathbb{R} \) such that \( x^n < 0 \). 2. Let \( n \) be an odd natural number, and suppose that for all \( x \in \mathbb{R} \), \( x^n \geq 0 \). 3. Let \( n \) be an even natural number. Suppose that for all \( x \in \mathbb{R} \), \( x^n < 0 \). 4. Take an arbitrary \( n \in \mathbb{N} \). Suppose there exists an \( x \in \mathbb{R} \) such that \( x^n < 0 \). 5. Let \( n \) be any odd natural number.
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