Introduction to Relations

Step 1: All self-pairs are present, so the relation is reflexive. Step 2: ((1,2)) and ((2,1)) are both present, and the needed transitive checks are satisfied. Step 3: When all three properties hold, the relation is an equivalence relation.

Step 1: All self-pairs are present, so the relation is reflexive. Step 2: ((1,2)) and ((2,3)) require ((1,3)), which is present. Step 3: ((1,2)) has no reverse ((2,1)), so it is not symmetric.

Step 1: Every element is related only to itself, so it is the identity relation. Step 2: The identity relation is reflexive, symmetric, and transitive. Step 3: Hence it is also an equivalence relation.

Step 1: Reflexivity needs a self-pair for every element. Step 2: If even one element has no self-pair, the condition fails. Step 3: When checking reflexivity, inspect every element of the whole set.

Step 1: (A\times A) has (4^2=16) pairs. Step 2: Reflexivity makes (4) self-pairs compulsory, leaving (12) pairs free. Step 3: Therefore, the number of reflexive relations is (2^{12}).

Step 1: In a relation from (A) to (B), the first entry must come from (A) and the second from (B). Step 2: ((1,3)) and ((2,4)) follow this rule. Step 3: Changing the order in ordered pairs changes the relation.

Step 1: Reflexivity requires the self-pair of every element. Step 2: For (A={1,2}), ((1,1)) and ((2,2)) are required. Step 3: Pairs with different elements do not complete reflexivity.

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.