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

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Question: If R is a relation on the set {1, 2, 3} defined by R = {(1, 1), (2, 2), (3, 3), (1, 2), (2, 1)}, which of the following is true?

Options:

Correct Answer: Both 1 and 2

Solution:

R is reflexive and symmetric, but not transitive. Thus, both 1 and 2 are true.

If R is a relation on the set {1, 2, 3} defined by R = {(1, 1), (2, 2), (3, 3), (1, 2), (2, 1)}, which of the following is true?

If R is a relation on the set {1, 2, 3} defined by R = {(1, 1), (2, 2), (3, 3), (1, 2), (2, 1)}, which of the following is true?

Step 1: Understand the set and relation. The set is {1, 2, 3} and the relation R consists of pairs: {(1, 1), (2, 2), (3, 3), (1, 2), (2, 1)}.
Step 2: Check if R is reflexive. A relation is reflexive if every element in the set is related to itself. Here, (1, 1), (2, 2), and (3, 3) are in R, so R is reflexive.
Step 3: Check if R is symmetric. A relation is symmetric if for every (a, b) in R, (b, a) is also in R. We have (1, 2) in R and (2, 1) in R, so R is symmetric.
Step 4: Check if R is transitive. A relation is transitive if whenever (a, b) and (b, c) are in R, then (a, c) must also be in R. Here, we have (1, 2) and (2, 1), but (1, 1) is already in R. However, (2, 1) and (1, 2) do not lead to (2, 2) being a new pair, so R is not transitive.
Step 5: Summarize the findings. R is reflexive (true), symmetric (true), and not transitive (false). Therefore, both statements about reflexivity and symmetry are true.

Reflexive Relation – A relation R on a set is reflexive if every element is related to itself, meaning (a, a) is in R for all a in the set.
Symmetric Relation – A relation R is symmetric if for every (a, b) in R, the pair (b, a) is also in R.
Transitive Relation – A relation R is transitive if whenever (a, b) and (b, c) are in R, then (a, c) must also be in R.