If R is a relation on the set {1, 2, 3, 4} 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, 4} defined by R = {(1, 1), (2, 2), (3, 3), (4, 4), (1, 2), (2, 1)}, what type of relation is R?

Options:

Correct Answer: Both reflexive and symmetric

Solution:

R is reflexive because it contains all pairs (a, a) and symmetric because (1,2) implies (2,1).

If R is a relation on the set {1, 2, 3, 4} defined by R = {(1, 1), (2, 2), (3, 3), (4, 4), (1, 2), (2, 1)}, what type of relation is R?

If R is a relation on the set {1, 2, 3, 4} defined by R = {(1, 1), (2, 2), (3, 3), (4, 4), (1, 2), (2, 1)}, what type of relation is R?

Step 1: Identify the set we are working with, which is {1, 2, 3, 4}.
Step 2: Look at the relation R, which is defined as R = {(1, 1), (2, 2), (3, 3), (4, 4), (1, 2), (2, 1)}.
Step 3: Check if R is reflexive. A relation is reflexive if it contains all pairs (a, a) for every element a in the set. Here, we have (1, 1), (2, 2), (3, 3), and (4, 4), which means R is reflexive.
Step 4: Check if R is symmetric. A relation is symmetric if for every pair (a, b) in R, the pair (b, a) is also in R. We see (1, 2) is in R, and (2, 1) is also in R, so R is symmetric.
Step 5: Conclude that R is both reflexive and symmetric.

Reflexive Relation – A relation R on a set is reflexive if every element is related to itself, meaning for every a in the set, (a, a) is in R.
Symmetric Relation – A relation R is symmetric if for every (a, b) in R, the pair (b, a) is also in R.