Question

If x is rational and y is irrational then x – y is​

Answer 1

Answer:

If x is rational and y is irrational, then xy is irrational. ... This implies that y is rational, a contradiction. We conclude that xy must be rational.

Step-by-step explanation:

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Answer 2

Topic

• Proof by contradiction.

Solution

① Proof by contradiction.

Suppose $x-y$ is rational.

A rational number is equal to $x-y$.

$\implies \text{Rational Number}=x-y$

$\implies \text{Rational Number}-x=-y$

The left-hand side is rational because rational numbers are closed under four operations. However, the right-hand side is irrational. Due to the result, the assumption is wrong and hence $x-y$ is irrational.

② Conclusion.

If $x$ is rational and $y$ is irrational, $x-y$ is irrational.

This is the required answer.

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Then why are rational numbers closed under four operations?

① Addition

$\implies \dfrac{a}{b} +\dfrac{c}{d} =\dfrac{ad+bc}{bd}$

② Subtraction

$\implies \dfrac{a}{b} -\dfrac{c}{d} =\dfrac{ad-bc}{bd}$

③ Multiplication

$\implies \dfrac{a}{b} \times \dfrac{c}{d} =\dfrac{ac}{bd}$

④ Division

$\implies \dfrac{a}{b} \div \dfrac{c}{d} =\dfrac{ad}{bc}$

All four numbers are rational since both denominators and numerators are integers. Integers are closed under three operations, except for division.