Introduction to Relations

Step 1: Symmetry needs the reverse of every existing pair. Step 2: The empty relation has no pairs, so the condition is not violated. Step 3: The empty relation can be symmetric, but it is not reflexive on a non-empty set.

Step 1: Transitivity fails only when two pairs exist but the required third pair is missing. Step 2: In the empty relation, no such two pairs exist, so the condition is not broken. Step 3: The empty relation is considered transitive.

Step 1: (A\times A) has (1^2=1) pair. Step 2: This one pair has two subsets: the empty subset and the subset containing that pair. Step 3: Use (2^{n^2}) for total relations.

Step 1: In (A\times B), there are (3) choices for the first entry and (2) choices for the second. Step 2: Total pairs are (3\times2=6). Step 3: In Cartesian product, multiply the numbers of elements.

Step 1: (A\times A) must contain (9) pairs. Step 2: In the given relation, only ((2,3)) and ((3,2)) are missing. Step 3: To make a relation universal, add every missing pair.

Step 1: Every number has the same parity as itself, so reflexivity holds. Step 2: If (a) has the same parity as (b), then (b) has the same parity as (a). Step 3: Same parity continues through a chain, so it is an equivalence relation.

Step 1: In an inverse relation, every pair is reversed. Step 2: In the given relation, the reverse of every pair is already present. Step 3: The inverse of such a symmetric relation is the relation itself.

Step 1: Reflexivity requires a self-pair for every element of the set. Step 2: (3) is also an element of (A), but ((3,3)) is missing. Step 3: Always look carefully at the original set before judging the relation.

Step 1: In transitivity, the second entry of the first pair and the first entry of the second pair must match. Step 2: In ((1,2)) and ((2,3)), the middle element (2) matches, so ((1,3)) is needed. Step 3: Identifying the correct chain is the key to transitivity questions.

Step 1: All self-pairs are present, so the relation is reflexive. Step 2: ((2,3)) has ((3,2)), and self-pairs reverse to themselves. Step 3: Reflexivity and symmetry should be checked separately.

Step 1: Reflexivity needs all self-pairs, and all four self-pairs are present. Step 2: The non-self pair ((1,2)) has its reverse ((2,1)). Step 3: In such questions, check self-pairs first and reverse pairs next.

Step 1: Symmetry requires the reverse of every pair. Step 2: ((1,2)) is present but ((2,1)) is absent, so symmetry fails. Step 3: Missing even one reverse pair makes the relation non-symmetric.

Step 1: A relation on a set with (n) elements is a subset of (A\times A). Step 2: Here (A\times A) has (5^2=25) pairs. Step 3: Therefore, the total number of relations is (2^{25}).

Step 1: (A\times A) has (4^2=16) pairs. Step 2: A reflexive relation must contain (4) self-pairs, leaving (12) pairs free. Step 3: The number is (2^{12}).

Step 1: All three self-pairs are present, so the relation is reflexive. Step 2: ((1,3)) is present but ((3,1)) is not. Step 3: Therefore, symmetry is definitely absent.

Step 1: From ((1,2)) and ((2,3)), transitivity requires ((1,3)). Step 2: This pair is missing from the relation. Step 3: In a long chain, start with adjacent connected pairs.

Step 1: From ((2,3)) and ((3,4)), transitivity requires ((2,4)). Step 2: ((2,4)) is missing, so transitivity is not complete. Step 3: Check all possible chains, not only the first one.

Step 1: (1) and (2) are related because ((1,2)) and ((2,1)) are present. Step 2: (3) is related only to itself. Step 3: Hence the classes are ({1,2}) and ({3}).

Step 1: Dividing (7) by (5) gives remainder (2). Step 2: In this relation, numbers with the same remainder lie in the same class. Step 3: In remainder-based relations, identify classes by the remainder.

Step 1: Every number has the same parity as itself, so reflexivity holds. Step 2: Same parity remains true when the order is reversed, so symmetry holds. Step 3: Same parity continues through a chain, so transitivity holds.