Relation \( R \) in the set \( A=\{1,2,3,4,\ldots\ldots,16\} \) defined as \( R=\{(x,y):3x=y,\ x,y\in A\} \), then which of the following are correct ?
A. domain of \( R=\{1,2,3,4,5\} \).
B. range of \( R=\{3,6,9,12,15\} \).
C. \( R \) is reflexive.
D. \( R \) is not symmetric.
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → A,B and D only
Since y = 3x must also lie in A, 3x ≤ 16, that is, x ≤ 5.
\( \displaystyle R=\{(1,3),(2,6),(3,9),(4,12),(5,15)\} \)
So the domain of R is {1, 2, 3, 4, 5}, the range is {3, 6, 9, 12, 15}, and no pair of R has equal entries.
Checking the statements
(A) domain of \( R=\{1,2,3,4,5\} \). — Correct. These are the first elements of the pairs.
(B) range of \( R=\{3,6,9,12,15\} \). — Correct. These are the second elements of the pairs.
(C) \( R \) is reflexive. — Incorrect. (1, 1) ∉ R, since 3(1) ≠ 1.
(D) \( R \) is not symmetric. — Correct. (1, 3) ∈ R but (3, 1) ∉ R, since 3(3) = 9 ≠ 1.