Can a relation be both symmetric and antisymmetric simultaneously?
ANo — they are logical opposites and cannot both hold for the same relation
BYes — if the only pairs (x, y) in the relation satisfy x = y, both properties hold trivially
CYes, but only for relations on infinite sets
DNo — antisymmetry explicitly negates symmetry by requiring asymmetric behavior
Symmetric means: if (x,y) ∈ R then (y,x) ∈ R. Antisymmetric means: if (x,y) ∈ R and (y,x) ∈ R then x = y. These are not logical opposites. The equality relation {(x,x) : x ∈ A} satisfies both: it is symmetric (trivially — (x,x) reversed is (x,x)) and antisymmetric (the only mutual pairs are (x,x) pairs where x = x). A relation fails to be antisymmetric only if there exist distinct x ≠ y with both (x,y) and (y,x) in R.
Question 2 Multiple Choice
Consider the 'divides' relation on positive integers: x divides y means y = kx for some positive integer k. Which combination of properties does this relation satisfy?
AReflexive, symmetric, and transitive — making it an equivalence relation
BReflexive, antisymmetric, and transitive — making it a partial order
CSymmetric and transitive, but not reflexive
DReflexive and symmetric, but not transitive
Divisibility is reflexive (x divides x, since x = 1·x), antisymmetric (if x|y and y|x for positive integers then x = y), and transitive (if x|y and y|z then x|z). It is NOT symmetric: 2 divides 6, but 6 does not divide 2. These three properties — reflexive, antisymmetric, transitive — define a partial order. Divisibility on ℕ is the canonical example of a partial order that is not a total order (2 and 3 are incomparable: neither divides the other).
Question 3 True / False
The 'is equal to' relation on any set is simultaneously reflexive, symmetric, transitive, AND antisymmetric.
TTrue
FFalse
Answer: True
Equality satisfies all four properties. Reflexive: x = x. Symmetric: if x = y then y = x. Transitive: if x = y and y = z then x = z. Antisymmetric: if x = y and y = x then x = y (trivially true). This means equality is both an equivalence relation (reflexive + symmetric + transitive) and satisfies antisymmetry. It is the only equivalence relation that is also a partial order.
Question 4 True / False
Most transitive relation is also reflexive.
TTrue
FFalse
Answer: False
Counterexample: the 'strictly less than' relation < on ℝ is transitive (if x < y and y < z then x < z) but not reflexive (no number is strictly less than itself). Another counterexample: the empty relation on any set is vacuously transitive but not reflexive. Transitivity says nothing about whether elements relate to themselves — it only constrains chains of distinct related pairs.
Question 5 Short Answer
A student checks whether the 'is a parent of' relation on people is an equivalence relation. Identify which required properties it fails, and explain why for each.
Think about your answer, then reveal below.
Model answer: It fails all three properties required for equivalence. Not reflexive: no one is their own parent. Not symmetric: if A is a parent of B, then B is not a parent of A (they are a child of A). Not transitive: if A is a parent of B and B is a parent of C, then A is a grandparent of C — not a parent. So 'is a parent of' is neither reflexive, symmetric, nor transitive.
Checking properties systematically — rather than relying on intuition — is the core skill. 'Is a sibling of' is symmetric but not reflexive or transitive. 'Is an ancestor of' is transitive but not reflexive or symmetric. 'Is a parent of' fails all three. Each failure corresponds to a concrete counterexample, which is how you prove a relation lacks a property.