Which of the following facts are not true? (Where \( R \) is set of real numbers)
(A) \( |x+y|\le|x|+|y|,\ x,y\in R \)
(B) \( |x-y|\ge|x|+|y|,\ x,y\in R \)
(C) \( |x+y|\ge|x|+|y|,\ x,y\in R \)
(D) \( |x-y|\le|x|+|y|,\ x,y\in R \)
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → (B) and (C) only
The triangle inequality \( |a+b|\le|a|+|b| \) holds for all real a, b. Putting b = −y also gives \( |x-y|\le|x|+|y| \). Each “≥” version fails for some x, y; one counter-example is enough to show that a “for all x, y” statement is false.
With x = y = 1:
\( \displaystyle |x-y|=0,\quad |x|+|y|=2 \)
With x = 1, y = −1:
\( \displaystyle |x+y|=0,\quad |x|+|y|=2 \)
So the two “≤” inequalities hold for all real x, y, while the two “≥” inequalities fail, since 0 is less than 2 in the two examples.
Checking the statements
(A) \( |x+y|\le|x|+|y|,\ x,y\in R \) — Correct. This is the triangle inequality.
(B) \( |x-y|\ge|x|+|y|,\ x,y\in R \) — Incorrect. For x = y = 1, |x − y| = 0 while |x| + |y| = 2.
(C) \( |x+y|\ge|x|+|y|,\ x,y\in R \) — Incorrect. For x = 1, y = −1, |x + y| = 0 while |x| + |y| = 2.
(D) \( |x-y|\le|x|+|y|,\ x,y\in R \) — Correct. It is the triangle inequality with y replaced by −y.
The question asks for the facts that are not true, so the answer is the statements marked Incorrect: (B) and (C).