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?

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Q: 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?

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

Steps: 5

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.