If R is a relation on the set {a, b, c} defined by R = {(a, b), (b, c)}, what can be said about R?

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Q: If R is a relation on the set {a, b, c} defined by R = {(a, b), (b, c)}, what can be said about R?

Solution: R is neither reflexive, symmetric, nor transitive as it does not satisfy any of the properties.

Steps: 7

Step 1: Understand what a relation is. A relation R on a set is a collection of ordered pairs from that set.
Step 2: Identify the set we are working with. The set is {a, b, c}.
Step 3: Look at the relation R given, which is R = {(a, b), (b, c)}.
Step 4: Check if R is reflexive. A relation is reflexive if every element is related to itself. Here, (a, a), (b, b), and (c, c) are not in R, so R is not reflexive.
Step 5: Check if R is symmetric. A relation is symmetric if for every (x, y) in R, (y, x) is also in R. Here, (b, a) and (c, b) are not in R, so R is not symmetric.
Step 6: Check if R is transitive. A relation is transitive if whenever (x, y) and (y, z) are in R, then (x, z) must also be in R. Here, we have (a, b) and (b, c), but (a, c) is not in R, so R is not transitive.
Step 7: Conclude that R is neither reflexive, symmetric, nor transitive.