2. For each of the following statements, determine whether the statement is true

Question

2. For each of the following statements, determine whether the statement is true or false. Prove the true statements, and for the false statements, write down their negations and prove the Note that you are allowed to use 2 as an irrational number without clarification. However, for any irrational numbers other than V2, you must prove that they are irrational. (a) If a and b are non-zero integers and r is an irrational number then abr is irrational (b) There exist real numbers z and y so that r and y are irrational, but ry is rational. (c) There exist real numbers and y so that r and y are irrational, and ry is also irrational. (d) For all positive real numbers r, if r is irrational then r is irrational.

Explanation / Answer

(a) If a and b are non-zero integers and x is an irrational number then a + bx is irrational.

This statement is true.

Proof: Let us assume the statement is false i.e a + bx is rational.

Let a + bx = p/q, where p and q are integers and q 0.

=> bx = p/q - a

=> bx = (p - aq) / q

=> x = (p - aq) / (qb)

=> Since p,q,a,b are all integers, so are p - aq and qb.

Let m = p - aq and n = qb.

=> x = m / n

=> x is rational which is a contradiction.

(b) There exist real numbers x and y so that x and y are irrational, but xy is rational.

This statement is true.

Proof: We know that 2 is irrational.

Let x = y = 2

=> xy = 2 * 2 = 2 which is rational.

Thus there exist irrational numbers x = 2 and y = 2 such that xy = 2 is rational.

(c) There exist real numbers x and y so that x and y are irrational, and xy is is also irrational.

The statement is true.

We know that 2 is irrational and by the result of (a), 1 + 2 is rational.

Multiplying => 2 (1 + 2) = 2 + 2 = 2 + 2

By the result of (a), this is rational.

Thus there exist two irrational numbers 2 and 1 + 2 such that 2 (1 + 2) is also irrational.

(d) For all positive real numbers x, if x is irrational, then x is irrational.

This statement is true.

Proof: Let the statement be false i.e let x be rational.

Let x = p/q where p and q are integers and q 0.

=> x = p2 / q2

Since p and q are integers, so are p2 and q2.

Let m = p2 and n = q2 where m and n are integers and n 0.

=> x = m / n

=> x is rational which is a contradiction.

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