Question

Relation \(R\) in the set \(A = \{1,2,3,4,\ldots,16\}\) 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:

• A. A and B only
• B. B and D only
• C. A,B and C only
• D. A,B and D only

Hint:

List the pairs: \(y=3x\) with \(y \leq 16\) gives \(x=1,\ldots,5\). Then test reflexive and symmetric.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
A relation \(R\) on a set is a set of ordered pairs. The domain is the set of first entries and the range is the set of second entries. Reflexive means \((x,x) \in R\) for every \(x\). Symmetric means \((x,y) \in R\) gives \((y,x) \in R\).

Step 2: List the pairs of R.
We need \(y=3x\) with both \(x\) and \(y\) in \(\{1,\ldots,16\}\). So \(3x \leq 16\), which gives \(x \leq 5\).
\[ R=\{(1,3),(2,6),(3,9),(4,12),(5,15)\} \]

Step 3: Check statement A.
The first entries are \(1,2,3,4,5\). So the domain is \(\{1,2,3,4,5\}\). Statement A is TRUE.

Step 4: Check statement B.
The second entries are \(3,6,9,12,15\). So the range is \(\{3,6,9,12,15\}\). Statement B is TRUE.

Step 5: Check statement C.
For \(R\) to be reflexive, we need \((1,1)\) in \(R\). But \(3 \times 1 = 3 \neq 1\). So \((1,1) \notin R\). Statement C is FALSE.

Step 6: Check statement D.
We have \((1,3) \in R\). For symmetry, we need \((3,1) \in R\), that is \(3 \times 3 = 1\), which is false. So \(R\) is not symmetric. Statement D is TRUE.

Step 7: Match with the options.
The true statements are A, B and D. That is option 4. Option 1 misses D. Option 2 misses A. Option 3 includes the false statement C.

Final Answer:
Statements A, B and D are correct. \[ \boxed{\text{Option 4: A, B and D only}} \]