Prove or disprove each of the following statements. A. If x and y are rational n

Question

Prove or disprove each of the following statements. A. If x and y are rational numbers, then x-y must also be a rational number. B. If x and y are rational numbers, then x and y must also be a rational number . Prove or disprove each of the following statements. A. If x and y are rational numbers, then x-y must also be a rational number. B. If x and y are rational numbers, then x and y must also be a rational number . A. If x and y are rational numbers, then x-y must also be a rational number. B. If x and y are rational numbers, then x and y must also be a rational number .

Explanation / Answer

Solution:

(A) A. If x and y are rational numbers, then x-y must also be a rational number.

Proof:

If x and y are rational,

then there exist integers a, b, c, d, with b 0, d 0,

such that x = a/b and y = c/d.

Then x - y = (ad - bc)/(bd).

Now,

ad - bc and bd are both integers

and, since b 0 and d 0,

we have bd 0.

Therefore x - y is rational.

(B) If x and y are rational numbers, then x and y ( x*y) must also be a rational number

Prove :

If x and y are rational,

then there exist integers a, b, c, d, with b 0, d 0,

such that x = a/b and y = c/d.

x*y = xy = ac/bd

ac is an integer

and

and, since b 0 and d 0,

we have bd 0.

therefore

xy is rational number

Related Questions

Navigate

• Browse (All)
• Browse P
• Subjects

---

---

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.