Introduction to Relations

Step 1: (A\times B) has (3\times2=6) ordered pairs. Step 2: Each subset forms a relation, so the total number of relations is (2^6). Step 3: In such questions first count the pairs in the Cartesian product.

Step 1: (A\times A) has (3^2=9) pairs. Step 2: For reflexivity, (3) diagonal pairs are fixed and the remaining (6) pairs are optional. Step 3: Hence the number is (2^6).

Step 1: The three diagonal pairs can be chosen or left independently. Step 2: The three off-diagonal reverse pairs can also be chosen or left. Step 3: There are (6) independent choices, so the number is (2^6).

Step 1: Reflexivity fixes all three diagonal pairs. Step 2: There are (3) off-diagonal reverse pairs, and each pair is optional. Step 3: Hence the total number of such relations is (2^3).

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, and adding two even differences gives an even difference. Step 3: Since all three properties hold, it is an equivalence relation.

Step 1: For every (a), (a\le a) is true, 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\le2) is true but (2\le1) is false, so it is not symmetric.

Step 1: (a<a) is never true, so the relation is not reflexive. Step 2: If (a<b) and (b<c), then (a<c), so it is transitive. Step 3: Relations based on less than are generally not symmetric.

Step 1: Every natural number divides itself, so the relation is reflexive. Step 2: If (a\mid b) and (b\mid c), then (a\mid c), so it is transitive. Step 3: (2\mid4) is true but (4\mid2) is false, so it is not symmetric.

Step 1: (a+a=2a) is always 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 carries through the chain, so it is transitive.

Step 1: (a-a=0) is divisible by (3), so the relation is reflexive. Step 2: If (a-b) is divisible by (3), then (b-a) is also divisible by (3). Step 3: Adding divisible differences again gives a divisible difference, so it is an equivalence relation.

Step 1: For every number, (|a|=|a|), so the relation is reflexive. Step 2: Equality remains true when reversed, so it is symmetric. Step 3: If two equalities connect, the third equality also follows, so it is transitive.

Step 1: (a^2=a^2) is true for every (a). Step 2: If (a^2=b^2), then (b^2=a^2) is also true. Step 3: A chain of equalities gives (a^2=c^2), so all three properties hold.

Step 1: For non-zero (a), (a^2>0), so the relation is reflexive. Step 2: If (ab>0), then (ba>0), so it is symmetric. Step 3: The relation means same sign, and same sign passes through a chain.

Step 1: (a-a=0), which is not greater than (0), so the relation is not reflexive. Step 2: If (a>b) and (b>c), then (a>c), so it is transitive. Step 3: Greater-than type relations are generally not symmetric.

Step 1: If (a+b=0), then (b+a=0), so the relation is symmetric. Step 2: (a+a=0) is not true for every real (a), so it is not reflexive. Step 3: Symmetry alone is not enough for equivalence.

Step 1: For symmetry, the reverse ((2,1)) of ((1,2)) must be added. Step 2: ((3,3)) is its own reverse, so no extra pair is needed for it. Step 3: For the smallest relation add only compulsory pairs.

Step 1: Reflexivity needs ((1,1),(2,2),(3,3)). Step 2: ((1,1)) is already present in the relation. Step 3: Therefore only ((2,2)) and ((3,3)) must be added.

Step 1: Transitivity requires ((1,3)) from ((1,2)) and ((2,3)). Step 2: ((1,3)) is not in the relation. Step 3: Hence the minimum pair to add is ((1,3)).

Step 1: In ((1,2)) and ((2,1)), the middle element (2) matches. Step 2: Transitivity requires ((1,1)). Step 3: Reverse pairs often create a need for a diagonal pair.

Step 1: From ((1,2)) and ((2,3)), ((1,3)) is needed. Step 2: Then ((1,3)) and ((3,4)) require ((1,4)). Step 3: In a transitive extension, also check requirements created by newly added pairs.

Step 1: In the inverse relation, the components of every pair are reversed. Step 2: Therefore the range of the original relation becomes the domain of the inverse. Step 3: Remember that domain and range interchange in the inverse.

Step 1: An equivalence relation needs reflexive, symmetric, and transitive properties. Step 2: Here reflexivity is missing, so the condition is not satisfied. Step 3: Two properties are not enough; all three must be confirmed.

Step 1: An equivalence relation is transitive. Step 2: From ((1,2)) and ((2,3)), transitivity gives ((1,3)). Step 3: In equivalence relations use the given chain to find required pairs.

Step 1: An equivalence relation is symmetric. Step 2: If ((4,7)) is present, symmetry gives ((7,4)). Step 3: In equivalence relations always remember the reverse pair.

Step 1: Same parity means both numbers are odd or both are even. Step 2: (1) is odd, and the odd numbers in ({1,2,3,4}) are (1,3). Step 3: In an equivalence class, write the elements of the same group.