Introduction to Relations

Step 1: A relation on a set is formed from a subset of its Cartesian product. Step 2: Here any subset of (A\times A) is a relation on (A). Step 3: In exams first identify the Cartesian product and then the relation.

Step 1: The number of ordered pairs in a Cartesian product is (n(A)\times n(B)). Step 2: Here (2\times2=4). Step 3: In such questions count the elements of both sets first.

Step 1: A relation on (A) is a subset of (A\times A). Step 2: (A\times A) has (2^2=4) pairs so the number of subsets is (2^4=16). Step 3: Remember (2^{n^2}) for total relations on a set with (n) elements.

Step 1: The universal relation contains all ordered pairs of the Cartesian product. Step 2: So on (A), the universal relation is (A\times A). Step 3: Do not confuse the universal relation with the empty relation.

Step 1: An empty relation has no ordered pair. Step 2: Hence it contains no pair at all. Step 3: Empty relation and universal relation are opposite style examples.

Step 1: The universal relation is (A\times A). Step 2: Since (n(A)=3), (A\times A) has (3^2=9) pairs. Step 3: Count all possible ordered pairs for a universal relation.

Step 1: In an identity relation every element is related to itself. Step 2: Therefore it contains pairs of the form ((a,a)). Step 3: Identify the identity relation through diagonal pairs.

Step 1: In an identity relation each element is related to itself. Step 2: For (1,2,3), the pairs are ((1,1),(2,2),(3,3)). Step 3: In such questions choose only pairs with equal components.

Step 1: In a reflexive relation every element must be related to itself. Step 2: So ((a,a)) must belong to (R) for each (a). Step 3: While checking reflexivity look for all diagonal pairs.

Step 1: For reflexivity, both ((1,1)) and ((2,2)) are needed. Step 2: Both diagonal pairs are present in the given relation, so it is reflexive. Step 3: Extra pairs do not spoil reflexivity.

Step 1: A reflexive relation must contain each element paired with itself. Step 2: Here ((3,3)) is missing, so the relation is not reflexive. Step 3: Check every element separately for reflexivity.

Step 1: In a symmetric relation, reversing a pair should still keep it in the relation. Step 2: So if ((a,b)) is present, ((b,a)) must also be present. Step 3: In symmetry questions quickly search for reverse pairs.

Step 1: The symmetric partner of a pair ((a,b)) is ((b,a)). Step 2: The reverse of ((1,2)) is ((2,1)). Step 3: In symmetric relations focus on the reversed ordered pair.

Step 1: For symmetry, every pair must have its reverse pair. Step 2: ((1,2)) is present but ((2,1)) is not, so the relation is not symmetric. Step 3: Symmetry depends on reverse pairs, not necessarily on diagonal pairs.

Step 1: In a transitive relation, two connected pairs should produce the third pair. Step 2: If ((a,b)) and ((b,c)) are present, ((a,c)) must be present. Step 3: For transitivity look for the common middle element.

Step 1: The transitive rule says ((a,b)) and ((b,c)) require ((a,c)). Step 2: From ((1,2)) and ((2,3)), the required pair is ((1,3)). Step 3: Remove the common middle element and join the first and last elements.

Step 1: An equivalence relation is a special type of relation. Step 2: It must be reflexive, symmetric, and transitive. Step 3: In exams check all three properties separately.

Step 1: It has both diagonal pairs, so it is reflexive. Step 2: Diagonal pairs reverse to themselves and transitivity also holds. Step 3: The identity relation is a good example of an equivalence relation.

Step 1: In a reflexive relation every element must be related to itself. Step 2: For element (2), ((2,2)) is missing, so the condition fails. Step 3: Missing even one diagonal pair destroys reflexivity.

Step 1: The smallest reflexive relation contains only the necessary diagonal pairs. Step 2: For three elements, the pairs are ((1,1),(2,2),(3,3)), so there are (3) pairs. Step 3: When the word smallest appears, count only compulsory pairs.

Step 1: The largest relation is the universal relation. Step 2: It contains all (3^2=9) pairs of (A\times A). Step 3: For the largest relation, take the full Cartesian product.

Step 1: (A\times A) has (n^2) pairs. Step 2: For (n=3), there are (9) pairs and the number of relations is (2^9). Step 3: Total relations are counted by the number of subsets.

Step 1: (A\times B) has (2\times3=6) ordered pairs. Step 2: A relation is a subset of (A\times B), so total relations are (2^6). Step 3: For two different sets, first count (mn) pairs.

Step 1: The domain contains the first components of ordered pairs. Step 2: Here the first components are (1) and (2). Step 3: To find the domain, look at the left side of each pair.

Step 1: The range contains the second components of ordered pairs. Step 2: Here the second components are (2,3,3), so the range is ({2,3}). Step 3: Write repeated elements only once in a set.