Introduction to Relations

Step 1: Reflexivity needs all self-pairs. Step 2: ((1,1)), ((2,2)), ((3,3)) are present and ((1,2)) has ((2,1)). Step 3: Check self-pairs first and then reverse pairs.

Step 1: An equivalence relation needs reflexivity, symmetry, and transitivity. Step 2: Here ((1,2)) is present but ((2,1)) is missing, so symmetry fails. Step 3: If even one property is missing, the relation is not equivalence.

Step 1: A relation on (A) is a subset of (A\times A). Step 2: (A\times A) has (4^2=16) pairs. Step 3: The number of subsets is (2^{16}), so that is the answer.

Option A is correct. A three-element set gives 3²=9 possible ordered pairs, so there are 2^9=512 total relations. A reflexive relation must contain the three diagonal pairs, while the remaining six pairs may be selected freely, giving 2^6=64 reflexive relations. Hence the number of non-reflexive relations is 512−64=448.

Step 1: Symmetry needs the reverse of every ordered pair. Step 2: The reverse of ((1,2)) is ((2,1)), which is not in the relation. Step 3: Missing even one reverse pair breaks symmetry.

Step 1: Transitivity needs ((a,c)) from ((a,b)) and ((b,c)). Step 2: From ((1,2)) and ((2,3)), ((1,3)) is required. Step 3: Take the first entry of the first pair and the second entry of the second pair.

Step 1: In the identity relation, every element is related only to itself. Step 2: So for (1), (2), and (3), the pairs are ((1,1)), ((2,2)), and ((3,3)). Step 3: Pairs with different entries are not part of the identity relation.

Step 1: A relation from (A) to (B) is a subset of (A\times B). Step 2: (A\times B) has (2\times3=6) pairs. Step 3: Therefore, the number of relations is (2^6).

Step 1: Since all self-pairs are present, the relation is reflexive. Step 2: ((1,2)) has ((2,1)), and ((2,3)) has ((3,2)). Step 3: Transitivity needs a separate check, so the definite properties here are reflexive and symmetric.

Step 1: Reflexivity requires a self-pair for every element. Step 2: ((1,1)), ((2,2)), and ((3,3)) are not in the relation. Step 3: A relation may look symmetric, but that does not replace reflexivity.

Step 1: The main chain ((1,2)) and ((2,3)) asks for ((1,3)). Step 2: ((1,3)) is present, and self-pairs do not create a problem. Step 3: Check all formed chains carefully for transitivity.

Step 1: All self-pairs are present, so reflexivity holds. Step 2: ((1,3)) is present but ((3,1)) is missing, so symmetry is absent. Step 3: For every non-self pair, always check its reverse pair.

Step 1: The universal relation is equal to (A\times A). Step 2: (A) has (3) elements, so there are (3^2=9) pairs. Step 3: In the universal relation, no possible pair is left out.

Step 1: Every element being related to itself identifies reflexivity. Step 2: The reverse-pair condition belongs to symmetry. Step 3: Remember the definitions of different properties separately.

Step 1: The given rule talks about the reverse pair of every pair. Step 2: This is the definition of a symmetric relation. Step 3: Translating symbols into words reduces mistakes.

Step 1: Here two connected pairs require a third pair. Step 2: This is the condition of a transitive relation. Step 3: Remember the chain ((a,b)), ((b,c)), and ((a,c)).

Step 1: A relation containing all pairs of (A\times A) is called the universal relation. Step 2: For (A={1,2,3}), there are (9) pairs. Step 3: If all possible pairs are present, the relation is universal.

Step 1: (\varnothing) contains no ordered pair. Step 2: So it is the empty subset of (A\times A). Step 3: Avoid confusing the empty relation with reflexive or universal relation.

Option B is correct. Reflexivity forces all three diagonal pairs to be included. For a symmetric relation, each of the three off-diagonal unordered pairs, namely {1,2}, {1,3}, and {2,3}, may either be included in both directions or omitted in both directions. These three independent choices give 2^3=8 relations. Thus exactly eight relations are both symmetric and reflexive.

Step 1: ((1,1)) and ((2,2)) make it reflexive. Step 2: Both ((1,2)) and ((2,1)) make it symmetric, and the full relation is transitive. Step 3: These three properties make it an equivalence relation.