If R is a relation on the set {x, y, z} defined by R = {(x, y), (y, z), (z, x)},

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Question: If R is a relation on the set {x, y, z} defined by R = {(x, y), (y, z), (z, x)}, what can be said about R?

Options:

Correct Answer: None of the above

Solution:

R is neither reflexive, symmetric, nor transitive.

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

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

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, which is {x, y, z}.
Step 3: Look at the relation R, which is given as R = {(x, y), (y, z), (z, x)}.
Step 4: Check if R is reflexive. A relation is reflexive if every element is related to itself. Here, (x, x), (y, y), and (z, z) are not in R, so R is not reflexive.
Step 5: Check if R is symmetric. A relation is symmetric if for every (a, b) in R, (b, a) is also in R. For example, (x, y) is in R, but (y, x) is not, so R is not symmetric.
Step 6: 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, (x, y) and (y, z) are in R, but (x, z) is not, so R is not transitive.
Step 7: Conclude that R is neither reflexive, symmetric, nor transitive.

Reflexive Relation – A relation R on a set is reflexive if every element is related to itself, i.e., 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, (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.