Introduction to Relations

Step 1: Symmetry needs the reverse of every pair. Step 2: The reverse pairs are ((2,1)), ((3,2)), and ((1,3)). Step 3: To make a relation symmetric, add the missing reverse pairs.

Step 1: The relation is reflexive and symmetric. Step 2: ((1,2)) and ((2,3)) require ((1,3)), and symmetry also needs ((3,1)). Step 3: In equivalence, transitive chains reveal missing pairs.

Step 1: (a=b) means an element is related only to itself. Step 2: So the pairs are ((1,1)), ((2,2)), and ((3,3)). Step 3: A relation based on equality is the identity relation.

Step 1: For every (a), (a\le a), so the relation is reflexive. Step 2: From (a\le b) and (b\le c), we get (a\le c), so it is transitive. Step 3: ((1,2)) is present but ((2,1)) is not, so it is not symmetric.

Step 1: (a<a) is never true, so the relation is not reflexive. Step 2: (1<2) and (2<3) imply (1<3), so it is transitive. Step 3: A relation going from smaller to larger is usually not symmetric.

Step 1: (a-a=0) is even, so the relation is reflexive. Step 2: If (a-b) is even, then (b-a) is also even, so it is symmetric. Step 3: Same parity continues through a chain, so it is transitive.

Step 1: (a-a=0) is divisible by (3), so it is reflexive. Step 2: If (a-b) is divisible, then (b-a) is also divisible, so it is symmetric. Step 3: Same-remainder divisibility also gives transitivity.

Step 1: In the inverse relation, the two entries of each pair are interchanged. Step 2: ((1,2)) becomes ((2,1)), ((2,3)) becomes ((3,2)), and ((3,3)) remains the same. Step 3: A self-pair does not change in the inverse.

Step 1: In the inverse relation, every pair is reversed. Step 2: If every reverse pair is already in the relation, the inverse remains the same. Step 3: This is the basic idea of symmetry.

Step 1: The relation contains ((1,1)), ((2,2)), and ((3,3)). Step 2: Therefore, every element is related to itself. Step 3: Extra pairs do not destroy reflexivity.

Step 1: Every distinct pair has its reverse in the relation. Step 2: ((1,2)) has ((2,1)), ((2,3)) has ((3,2)), and ((1,3)) has ((3,1)). Step 3: Complete presence of reverse pairs shows symmetry.

Step 1: Reflexivity applies to every element of the whole set. Step 2: Since (3) is in the set, ((3,3)) must be present. Step 3: Having only some self-pairs does not complete reflexivity.

Step 1: (A\times A) contains all possible pairs, so all self-pairs are present. Step 2: Every reverse pair and every pair needed for transitivity is also present. Step 3: The universal relation is often also an equivalence relation.

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.