Answer
true
Work Step by Step
The original statement "If r and s are any two rational numbers, then (r+s)/2 is rational" is indeed true.
Proof:
1. Let r and s be any two rational numbers.
2. By definition, a rational number can be expressed as the ratio of two integers, where the denominator is not zero.
3. Let r = a/b and s = c/d, where a, b, c, and d are integers, and b ≠ 0 and d ≠ 0.
4. The sum of two rational numbers is also a rational number, so r + s = (a/b) + (c/d) = (ad + bc)/(bd), which is a ratio of two integers with $bd\ne0$.
5. Dividing the sum by 2, we have (r + s)/2 = [(ad + bc)/(bd)]/2 = (ad + bc)/(2bd).
6. Since ad + bc and 2bd are integers (as the sum and product of integers), (r + s)/2 is expressed as the ratio of two integers with a non-zero denominator.
7. Therefore, (r + s)/2 is a rational number.
So, the original statement holds true.
