Analyzing True Mathematical Statements

We are asked to determine which of the given statements is true. Each statement presents a condition using an "if and only if" structure, which means both parts of the implication must hold true for the statement to be considered true.

Evaluating Statement 1: Rational Numbers and Squares

The first statement is: A number is rational if and only if its square is rational.

• Part 1: If a number \(x\) is rational, is \(x^2\) rational?

  Yes. A rational number can be written as a fraction \(\frac{p}{q}\) where \(p, q\) are integers and \(q \neq 0\). Its square is \((\frac{p}{q})^2 = \frac{p^2}{q^2}\), which is also a fraction of integers with a non-zero denominator, hence rational.

• Part 2: If the square of a number \(x^2\) is rational, is \(x\) rational?

  No. Consider the number \(x = \sqrt{2}\). We know \(\sqrt{2}\) is an irrational number. However, its square is \(x^2 = (\sqrt{2})^2 = 2\), which is a rational number. This provides a counterexample.

Since the second part of the "if and only if" statement is false, the entire statement is false.

Evaluating Statement 2: Odd Integers and n(n+1)

The second statement is: A number \(n\) is odd if and only if \(n(n + 1)\) is even.

• Part 1: If an integer \(n\) is odd, is \(n(n + 1)\) even?

  Yes. If \(n\) is odd, then \(n+1\) is the next integer, which must be even. The product of an odd number and an even number is always even.

• Part 2: If \(n(n + 1)\) is even, is the integer \(n\) odd?

  No. Consider the case where \(n\) is an even integer. If \(n\) is even, then \(n+1\) is odd. The product of an even number and an odd number is always even. For example, if \(n=4\), \(n(n+1) = 4 \times 5 = 20\), which is even, but \(n=4\) is not odd.

Since the second part of the "if and only if" statement is false, the entire statement is false.

Evaluating Statement 3: Odd Integers and n²+2n

The third statement is: An integer \(n\) is odd if and only if \(n^2 + 2n\) is odd.

Let's analyze the expression \(n^2 + 2n\). We can factor it as \(n(n + 2)\).

• Part 1: If an integer \(n\) is odd, is \(n^2 + 2n\) odd?

  Yes. If \(n\) is an odd integer, then \(n+2\) is also an odd integer (Odd + Even = Odd). The product of two odd integers is always odd. We can also write \(n = 2k+1\) for some integer \(k\). \[n^2 + 2n = (2k+1)^2 + 2(2k+1) = (4k^2 + 4k + 1) + (4k + 2) = 4k^2 + 8k + 3 = 2(2k^2 + 4k + 1) + 1\] This shows \(n^2 + 2n\) is of the form \(2m+1\), which is odd.

• Part 2: If \(n^2 + 2n\) is odd, is the integer \(n\) odd?

  Yes. We can prove this using the contrapositive: If \(n\) is not odd (i.e., \(n\) is even), then \(n^2 + 2n\) is not odd (i.e., \(n^2 + 2n\) is even). If \(n\) is even, we can write \(n=2k\) for some integer \(k\). \[n^2 + 2n = (2k)^2 + 2(2k) = 4k^2 + 4k = 2(2k^2 + 2k)\] This shows \(n^2 + 2n\) is of the form \(2m\), which is even. So, if \(n\) is even, \(n^2 + 2n\) is even. This means if \(n^2 + 2n\) is odd, \(n\) must be odd. The contrapositive is true, so this implication is true.

Since both parts of the "if and only if" statement are true, the entire statement is true.

Evaluating Statement 4: Irrational Numbers and Squares

The fourth statement is: A number is irrational if and only if its square is irrational.

• Part 1: If a number \(x\) is irrational, is \(x^2\) irrational?

  No. As shown in the analysis of Statement 1, \(x = \sqrt{2}\) is irrational, but \(x^2 = 2\) is rational.

• Part 2: If the square of a number \(x^2\) is irrational, is \(x\) irrational?

  Yes. We can prove this by contrapositive: If \(x\) is not irrational (i.e., \(x\) is rational), then \(x^2\) is not irrational (i.e., \(x^2\) is rational). As shown in the analysis of Statement 1, if \(x\) is rational, \(x^2\) is rational. The contrapositive is true, so this implication is true.

Since the first part of the "if and only if" statement is false, the entire statement is false.

Conclusion on True Statement

Based on the detailed analysis, only the statement "An integer \(n\) is odd if and only if \(n^2 + 2n\) is odd" is true.

Revision Table: Properties of Odd and Even Integers

Additional Information: Understanding "If and Only If"

The phrase "if and only if" is crucial in mathematics and logic. It indicates a strong relationship between two statements, say Statement P and Statement Q. "P if and only if Q" means that P being true is a necessary and sufficient condition for Q to be true, and vice versa. It combines two conditional statements:

1. If P, then Q (\(P \implies Q\))
2. If Q, then P (\(Q \implies P\))

Both of these implications must hold for the "if and only if" statement to be true. In our problem, we checked both directions for each given statement involving number properties.