Introduction to Relations

Step 1: In an identity relation, each element is paired only with itself. Step 2: For (3) elements, the pairs are ((1,1)), ((2,2)), and ((3,3)). Step 3: The number of identity pairs equals the number of elements.

Step 1: A universal relation needs all pairs of (A\times A). Step 2: For (A={1,2}), ((1,2)) and ((2,1)) are also needed, but they are absent. Step 3: Remember the difference between identity and universal relations.

Step 1: In (A\times A), there are (3) choices for the first position and (3) for the second position. Step 2: Total pairs are (3\times3=9). Step 3: In Cartesian product, multiply, do not add.

Step 1: (A\times A) has (2^2=4) pairs. Step 2: Every relation is a subset of these (4) pairs. Step 3: Therefore, the total number of relations is (2^4=16).

Step 1: Reflexivity needs self-pairs for both elements. Step 2: ((1,1)) is present, but ((2,2)) is missing. Step 3: One self-pair is not enough; all ((a,a)) pairs are required.

Step 1: A relation with no ordered pair is called an empty relation. Step 2: It is the empty subset of (A\times A). Step 3: Identify an empty relation by zero pairs.

Step 1: A relation containing all pairs of (A\times A) is called the universal relation. Step 2: No possible ordered pair is left out. Step 3: If all pairs are present, identify it as universal.

Step 1: Symmetry needs the reverse of every non-identical pair. Step 2: ((1,2)) has ((2,1)), and self-pairs reverse to themselves. Step 3: Pairs of the form ((a,a)) do not create a problem for symmetry.

Step 1: Symmetry requires the reverse of every pair. Step 2: The reverse of ((2,3)) is ((3,2)), which is not given. Step 3: Having some reverse pairs is not enough; all are required.

Step 1: Transitivity needs ((a,c)) from ((a,b)) and ((b,c)). Step 2: From ((1,2)) and ((2,3)), we get ((1,3)). Step 3: For transitivity, take the first element of the first pair and the second element of the second pair.

Step 1: The reverse of ((1,2)), namely ((2,1)), should be present. Step 2: Since ((2,1)) is missing, the relation is not symmetric. Step 3: Do not get confused by self-pairs; check reverse non-self pairs.

Step 1: In an identity relation, every element is related to itself. Step 2: Since (4) is an element of the set, ((4,4)) must be present. Step 3: Identity relation does not include pairs of different elements.

Step 1: A relation on (A) means both positions in the ordered pairs come from (A). Step 2: Therefore, all pairs come from (A\times A). Step 3: Understand the difference between a relation on (A) and a relation from (A) to (B).

Step 1: In (A\times B), there are (2) choices for the first position and (3) choices for the second. Step 2: Total pairs are (2\times3=6). Step 3: For different sets also, multiply to count Cartesian product pairs.

Step 1: All self-pairs are present, so the relation is reflexive. Step 2: The reverse of each self-pair is itself, and chains also remain valid. Step 3: Identity relation is a common example of a reflexive, symmetric, and transitive relation.

Step 1: An equivalence relation needs three properties. Step 2: These properties are reflexivity, symmetry, and transitivity. Step 3: To identify an equivalence relation, check all three conditions separately.

Step 1: An equivalence relation requires all three properties together. Step 2: Since transitivity is absent, the condition is not complete. Step 3: Missing even one property prevents a relation from being an equivalence relation.

Step 1: In a relation on (A), both entries must come from (A). Step 2: In ((3,1)), the element (3) is not in (A). Step 3: While checking a pair, first see whether its entries belong to the given set.

Step 1: In an identity relation, each element is related only to itself. Step 2: For (x), we need ((x,x)), and for (y), we need ((y,y)). Step 3: Pairs with different elements are not part of the identity relation.

Step 1: Transitivity needs ((a,c)) from ((a,b)) and ((b,c)). Step 2: Here ((1,2)) and ((2,2)) again require ((1,2)). Step 3: Sometimes the required pair is already one of the given pairs.