Types of relations

Correct answer: A. yes

Step 1: The universal relation has all pairs, so all diagonal pairs are present. Step 2: The reverse of every pair is also present. Step 3: The pair required for transitivity is also present, so it is an equivalence relation.

Correct answer: B. no

Step 1: A non-empty set has at least one element. Step 2: Reflexivity needs the diagonal pair of that element. Step 3: The empty relation has no pair, so it is not reflexive.

Correct answer: A. yes

Step 1: To fail symmetry, there must be a pair whose reverse is absent. Step 2: The empty relation has no pair at all. Step 3: Therefore no counterexample exists, and it is symmetric.

Correct answer: A. yes

Step 1: To fail transitivity, two connected pairs are needed. Step 2: The empty relation has no pair, so such a chain never forms. Step 3: Hence the empty relation is considered transitive.

Correct answer: C. (3)

Step 1: In the identity relation, each element is paired only with itself. Step 2: The set (A) has (3) elements. Step 3: Therefore the identity relation has (3) diagonal pairs.

Correct answer: C. (9)

Step 1: The universal relation is (A\times A). Step 2: If (A) has (3) elements, (A\times A) has (3\times3=9) pairs. Step 3: For a universal relation, count the size of the Cartesian product.

Correct answer: A. (4)

Step 1: The identity relation contains only diagonal pairs. Step 2: Each element gives one diagonal pair. Step 3: On (4) elements, the identity relation has (4) pairs.

Correct answer: D. (16)

Step 1: The universal relation is (A\times A). Step 2: For (4) elements, (A\times A) contains (4^2=16) pairs. Step 3: A universal relation counts all possible ordered pairs.

Correct answer: A. identity relation

Step 1: The identity relation contains only pairs of the form ((a,a)). Step 2: The reverse of such a pair is the same pair. Step 3: Therefore the identity relation is always symmetric.

Correct answer: B. identity relation

Step 1: Reflexivity requires all pairs ((a,a)). Step 2: The identity relation contains exactly all these diagonal pairs. Step 3: Hence the identity relation is always reflexive.

Correct answer: A. identity relation

Step 1: The identity relation contains only pairs with equal elements. Step 2: From ((a,a)) and ((a,a)), transitivity requires ((a,a)), which is present. Step 3: Therefore the identity relation is transitive.

Correct answer: A. equivalence relation

Step 1: Look at the three properties together. Step 2: A relation that is reflexive, symmetric and transitive is called an equivalence relation. Step 3: Remembering the list of properties is the easiest method.

Correct answer: B. partial order relation

Step 1: A partial order relation is based on comparison. Step 2: It has reflexive, antisymmetric and transitive properties. Step 3: When you see antisymmetry with reflexivity and transitivity, think of partial order.

Correct answer: A. both ((a,b)) and ((b,a)) occur together

Step 1: Antisymmetry prevents two-way relation between distinct elements. Step 2: If (a\ne b), both ((a,b)) and ((b,a)) cannot occur together. Step 3: Diagonal pairs do not violate antisymmetry.

Correct answer: A. diagonal pair ((a,a))

Step 1: In an irreflexive relation, no element is related to itself. Step 2: So ((a,a)) must not be present for any (a). Step 3: Off-diagonal pairs may still be present in an irreflexive relation.

Correct answer: A. yes

Step 1: An irreflexive relation must have no diagonal pair. Step 2: The given relation contains none of ((1,1),(2,2),(3,3)). Step 3: Therefore it is irreflexive.

Correct answer: A. because ((1,1)) is present

Step 1: An irreflexive relation must not contain any pair of the form ((a,a)). Step 2: Here ((1,1)) is present. Step 3: Even one diagonal pair is enough to fail irreflexivity.

Correct answer: A. equivalence relation

Step 1: All diagonal pairs are present, so it is reflexive. Step 2: ((1,2)) is accompanied by ((2,1)), so it is symmetric. Step 3: The required transitive pairs are also present, so it is an equivalence relation.

Correct answer: A. partial order relation

Step 1: Every natural number divides itself, so it is reflexive. Step 2: If (a\mid b) and (b\mid a), then (a=b), so it is antisymmetric. Step 3: Divisibility passes through a middle element, so it is transitive and hence a partial order.

Correct answer: A. it is reflexive and transitive but not symmetric

Step 1: (a\le a), so the relation is reflexive. Step 2: If (a\le b) and (b\le c), then (a\le c), so it is transitive. Step 3: (2\le3) is true but (3\le2) is false, so it is not symmetric.

Correct answer: A. transitive

Step 1: From ((1,2)) and ((2,3)), ((1,3)) is required. Step 2: ((1,3)) is present in the relation. Step 3: Hence the relation satisfies transitivity.

Correct answer: A. no two-way off-diagonal pair

Step 1: Symmetry asks for the reverse pair. Step 2: Antisymmetry does not allow both directions for distinct elements. Step 3: Therefore no two-way off-diagonal pair can occur between distinct elements.

Correct answer: A. reflexive and symmetric relation

Step 1: Every element being related to itself means reflexivity. Step 2: Having the reverse of every related pair means symmetry. Step 3: It becomes an equivalence relation only after transitivity is also proved.

Correct answer: A. ({(a,a),(b,b)})

Step 1: In the identity relation, each element is related only to itself. Step 2: For (a) and (b), the diagonal pairs are ((a,a)) and ((b,b)). Step 3: Hence the correct relation is ({(a,a),(b,b)}).

Correct answer: A. ((1,2))

Step 1: The universal relation contains all pairs of (A\times A). Step 2: Since (A) has only (1) and (2), ((1,2)) is a possible pair. Step 3: Pairs containing (3) are not in (A\times A).