If R is a reation on the set `A={1,2,3}` given by `R={(1,1),(2,2)(1,3)}` then R is

Options

A. reflexive
B. symmetric
C. transitive
D. none of these

Answer

To determine the type of relation R on the set A = {1, 2, 3}, where R = {(1, 1), (2, 2), (1, 3)}, we will check if R is reflexive, symmetric, or transitive.

### Step 1: Check for Reflexivity
A relation R is reflexive if every element in the set A is related to itself. This means that for every element a in A, the pair (a, a) must be in R.

- For A = {1, 2, 3}, we need to check for (1, 1), (2, 2), and (3, 3).
- We see that (1, 1) and (2, 2) are present in R, but (3, 3) is not present.

**Conclusion:** R is not reflexive.

### Step 2: Check for Symmetry
A relation R is symmetric if whenever (a, b) is in R, then (b, a) must also be in R.

- In R, we have (1, 3). For R to be symmetric, (3, 1) must also be in R, but it is not present.

**Conclusion:** R is not symmetric.

### Step 3: Check for Transitivity
A relation R is transitive if whenever (a, b) and (b, c) are in R, then (a, c) must also be in R.

- In R, we check for pairs:
- We have (1, 3) but there is no (3, b) for any b in R.
- Since we do not have any pairs that can form a transitive condition, we cannot conclude transitivity.

**Conclusion:** R is not transitive.

### Final Conclusion
Since R is neither reflexive, nor symmetric, nor transitive, we conclude that R is none of these types of relations.

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