Exercises for Question Type 3: Using contradiction with conditions - IB | RevisionDojo

Question Type 3: Using contradiction with conditions Exercises

Question 1

Let $x \ne 0$ be irrational. Prove by contradiction that $\dfrac{1}{x}$ is irrational.

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Question 2

Let $n \in \mathbb{Z}$ .

Prove by contradiction that if $n^2$ is even, then $n$ is even.

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Question 3

Given $x \notin \mathbb{Q}$ , prove by contradiction that $3 - 5x \notin \mathbb{Q}$ .

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Question 4

Let $x \notin \mathbb{Q}$ . Prove by contradiction that $|x| \notin \mathbb{Q}$ .

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Question 5

Let $a,b \in \mathbb{Q}$ with $a \ne 0$ . Given $x \notin \mathbb{Q}$ , prove by contradiction that $ax + b \notin \mathbb{Q}$ .

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Question 6

Let $x \notin \mathbb{Q}$ and $r \in \mathbb{Q}\setminus\{0\}$ . Prove by contradiction that $\frac{r}{x} + 5$ is irrational.

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Question 7

Given $x \notin \mathbb{Q}$ , prove by contradiction that $2x+1 \notin \mathbb{Q}$ .

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Question 8

Let $r ∈ \mathbb{Q}\setminus\{0\}$ . If $x \notin \mathbb{Q}$ , prove by contradiction that $\dfrac{x}{r} \notin \mathbb{Q}$ .

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