Introduction to Relations

Step 1: In a reflexive relation, every element must be related to itself. Step 2: Here both ((1,1)) and ((2,2)) are present. Step 3: In exams, first check all ((a,a)) pairs for reflexivity.

Step 1: An empty relation has no ordered pair. Step 2: The number of elements in the set does not matter for an empty relation. Step 3: Always distinguish empty relation from universal relation in exams.

Step 1: A universal relation contains all ordered pairs of (A\times A). Step 2: If (A) has (4) elements, then (A\times A) has (4^2=16) pairs. Step 3: For a universal relation, remember (n^2).

Step 1: In a symmetric relation, if ((a,b)) is present, then ((b,a)) must also be present. Step 2: Here both ((1,2)) and ((2,1)) are present. Step 3: In such questions, immediately look for the reverse pair.

Step 1: Reflexivity needs every element to be related to itself. Step 2: The set has (1) and (2), and both ((1,1)) and ((2,2)) are given. Step 3: Extra pairs do not destroy reflexivity.

Step 1: A symmetric relation must contain the reverse of every pair. Step 2: Here ((3,5)) is present but ((5,3)) is absent, so symmetry fails. Step 3: For symmetry, focus on reverse pairs.

Step 1: An identity relation contains only pairs of the form ((a,a)). Step 2: Here every element is related only to itself. Step 3: An identity relation is always reflexive.

Step 1: In (A\times A), each element can appear in the first and second positions. Step 2: For (2) elements, the total pairs are (2^2=4). Step 3: Before counting relations, first count (A\times A) correctly.

Step 1: A relation from (A) to (B) is a collection of ordered pairs. Step 2: These pairs are chosen from (A\times B). Step 3: For definition questions, remember that a relation is a subset of the Cartesian product.

Step 1: A relation on (A) is a subset of (A\times A). Step 2: (A\times A) has (3^2=9) pairs, so the number of subsets is (2^9). Step 3: For total relations, remember (2^{n^2}).

Step 1: Transitivity means if ((a,b)) and ((b,c)) are present, then ((a,c)) must also be present. Step 2: Here ((1,2)) and ((2,3)) are accompanied by ((1,3)). Step 3: In transitivity, always check the required third pair.

Step 1: A transitive relation must complete the forward chain. Step 2: From ((1,2)) and ((2,3)), the pair ((1,3)) is required, but it is missing. Step 3: In such questions, carefully check chain-like pairs.

Step 1: A universal relation contains all pairs of (A\times A). Step 2: For (A={1,2}), all four possible pairs are given. Step 3: If no possible pair is left out, the relation is universal.

Step 1: Reflexivity is about being related to oneself. Step 2: So for every element (a), the pair ((a,a)) must be in the relation. Step 3: In definition questions, connect the symbols with their meaning.

Step 1: Symmetry means the reverse ordered pair must also belong to the relation. Step 2: So ((b,a)) must be present whenever ((a,b)) is present. Step 3: Remember symmetry as the reverse-pair rule.

Step 1: Transitivity is the rule of completing a relation chain. Step 2: If ((a,b)) and ((b,c)) are present, then ((a,c)) must be present. Step 3: In transitivity, notice the common middle element.

Step 1: Symmetry requires the reverse pair. Step 2: The reverse of ((2,3)) is ((3,2)). Step 3: While reversing, interchange both positions.

Step 1: Reflexivity needs ((1,1)), ((2,2)), and ((3,3)). Step 2: All three pairs are present in the relation. Step 3: Extra pairs do not remove reflexivity.

Step 1: On (A={1,2}), reflexivity needs ((1,1)) and ((2,2)). Step 2: The relation has only ((1,2)), so the required self-pairs are missing. Step 3: In reflexivity, focus on self-pairs.

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.