Question

Consider the relation R defined on the set of positive integers as (x,y) ∈ R if x divides y. Choose all the true statements. R is reflexive. R is symmetric. R is antisymmetric. R is transitive. R is a partial order. R is a total order. R is an equivalence relation.

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R is reflexive: This is true. A relation R is reflexive if every element is related to itself. In this case, every positive integer x divides itself, so the relation is reflexive.
R is symmetric: This is false. A relation R is symmetric if for every pair (x, y) in R, the pair (y, x) is also in R. In this case, if x divides y, it does not necessarily mean that y divides x. For example, 2 divides 4, but 4 does not divide 2.
R is antisymmetric: This is true. A relation R is antisymmetric if whenever (x, y) and (y, x) are in R, then x = y. In this case, if x divides y and y divides x, it must be the case that x = y.
R is transitive: This is true. A relation R is transitive if whenever (x, y) and (y, z) are in R, then (x, z) is also in R. In this case, if x divides y and y divides z, then x divides z.
R is a partial order: This is true. A relation R is a partial order if it is reflexive, antisymmetric, and transitive. As we’ve established, all three of these properties hold for R.
R is a total order: This is false. A relation R is a total order if it is a partial order and, for all x and y, either x is related to y or y is related to x. In this case, there are pairs of positive integers where neither integer divides the other (for example, 2 and 3), so R is not a total order.
R is an equivalence relation: This is false. A relation R is an equivalence relation if it is reflexive, symmetric, and transitive. As we’ve established, R is not symmetric, so it cannot be an equivalence relation.

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