If R is a relation on set A = {1, 2, 3} defined by R = {(1, 1), (2, 2), (3, 3)},

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Question: If R is a relation on set A = {1, 2, 3} defined by R = {(1, 1), (2, 2), (3, 3)}, is R symmetric?

Options:

Correct Answer: Yes

Solution:

A relation is symmetric if for every (a, b) in R, (b, a) is also in R. Since R only contains pairs of the form (a, a), it is symmetric.

If R is a relation on set A = {1, 2, 3} defined by R = {(1, 1), (2, 2), (3, 3)}, is R symmetric?

If R is a relation on set A = {1, 2, 3} defined by R = {(1, 1), (2, 2), (3, 3)}, is R symmetric?

Correct Answer: Yes, R is symmetric.

Step 1: Understand what a relation is. A relation R on a set A is a collection of ordered pairs where the first element comes from set A and the second element also comes from set A.
Step 2: Identify the set A. In this case, A = {1, 2, 3}.
Step 3: Look at the relation R. Here, R = {(1, 1), (2, 2), (3, 3)}. This means R contains pairs where the first and second elements are the same.
Step 4: Define what it means for a relation to be symmetric. A relation R is symmetric if for every pair (a, b) in R, the pair (b, a) is also in R.
Step 5: Check the pairs in R. The pairs are (1, 1), (2, 2), and (3, 3). Each of these pairs is of the form (a, a).
Step 6: Verify symmetry. For each pair (a, a), the reverse pair (a, a) is also in R. For example, (1, 1) is in R, and its reverse (1, 1) is also in R.
Step 7: Conclude that since all pairs in R satisfy the symmetry condition, R is symmetric.

Symmetric Relation – A relation R is symmetric if for every pair (a, b) in R, the pair (b, a) is also in R.
Reflexive Relation – A relation is reflexive if every element is related to itself, which is the case here since R contains (1, 1), (2, 2), and (3, 3).