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Show that these statements about the real number $x$ are equivalent:

$x$ is irrational,

$3x+2$ is irrational,

$x/2$ is irrational.
in Mathematical Logic
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To show that the statements about the real number x are equivalent, we need to prove that if one statement is true, then the other two statements are also true. Additionally, we need to prove that if one statement is false, then the other two statements are also false.

1. Assume x is irrational.
To show that 3x + 2 is irrational, we will prove it by contradiction. Let's assume that 3x + 2 is rational. By definition, a rational number can be expressed as p/q, where p and q are integers and q is not equal to zero. So, we can write 3x + 2 = p/q, where p and q are integers and q ≠ 0.

Rearranging the equation, we have:
3x = p/q - 2
3x = (p - 2q)/q

Since p - 2q and q are integers, 3x should also be rational. However, this contradicts our assumption that x is irrational. Therefore, if x is irrational, 3x + 2 must also be irrational.

To show that x/2 is irrational, we will again prove it by contradiction. Let's assume that x/2 is rational. By definition, a rational number can be expressed as p/q, where p and q are integers and q is not equal to zero. So, we can write x/2 = p/q, where p and q are integers and q ≠ 0.

Rearranging the equation, we have:
x = (2p)/q

Since 2p and q are integers, x should also be rational. However, this contradicts our assumption that x is irrational. Therefore, if x is irrational, x/2 must also be irrational.

2. Assume x is rational.
If x is rational, we can express it as x = p/q, where p and q are integers and q ≠ 0.

Now let's consider 3x + 2:
3x + 2 = 3(p/q) + 2 = (3p + 2q)/q

Since 3p + 2q and q are integers, 3x + 2 is rational. Therefore, if x is rational, 3x + 2 is also rational.

Now let's consider x/2:
x/2 = (p/q)/2 = p/(2q)

Since p and 2q are integers, x/2 is rational. Therefore, if x is rational, x/2 is also rational.

By proving both directions, it is proven that the statements are equivalent. Hence, the statements "x is irrational," "3x + 2 is irrational," and "x/2 is irrational" are all equivalent.

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