Equivalence relation

Step 1: An equivalence relation is reflexive. Step 2: Due to reflexivity, for every (a\in A), ((a,a)\in R). Step 3: Self-relation is the first check for equivalence.

Step 1: Getting (bRa) from (aRb) shows reversal of order. Step 2: This is the main sign of the symmetric property. Step 3: Identify symmetry by noticing the reverse relation.

Step 1: Here a chain is formed from (a) to (b) and from (b) to (c). Step 2: The same chain gives the relation from (a) to (c). Step 3: Whenever a chain gives a direct relation, identify transitivity.

Step 1: All elements of the given set are divisible by (2). Step 2: Therefore, every element is related to every other element, so the relation is universal. Step 3: A universal relation is also an equivalence relation.

Step 1: All elements are odd, and the sum of two odd numbers is even. Step 2: Therefore, every pair belongs to the relation, making it universal. Step 3: If all pairs are present, all three equivalence properties hold.

Step 1: The condition (a=b) gives all identity pairs. Step 2: (1) and (2) form one class, and (3) forms a singleton class. Step 3: Identify the equivalence relation through the classes ({1,2}) and ({3}).

Step 1: An equivalence relation is reflexive. Step 2: Therefore, the identity pair of any element cannot be missing. Step 3: Catch missing identity pairs first.

Step 1: (1) is related to itself. Step 2: (1) is also related to (4), so the class of (1) is ({1,4}). Step 3: To find a class, write all elements related to the given element.

Step 1: Reflexivity quickly checks identity pairs. Step 2: Then use reverse pairs for symmetry and chains for transitivity. Step 3: This order reduces the chance of mistakes.

Step 1: Every triangle has the same area as itself, so the relation is reflexive. Step 2: Equality of area remains true when order is reversed, and it also passes through a chain. Step 3: For equality-based relations, check all three properties separately.

Step 1: Every element is related to itself, so the relation is reflexive. Step 2: The pair ((1,2)) has its reverse ((2,1)), so symmetry holds. Step 3: The classes are ({1,2}) and ({3}), so no extra pair is needed.

Step 1: Integers related to (7) must differ from (7) by a multiple of (4). Step 2: (-1,3,7,11) all have the same remainder (3) on division by (4). Step 3: For such questions, group all elements with the same remainder.

Step 1: The relation groups numbers by the same parity. Step 2: (1,3) are odd and (2,4) are even, so the two equivalence classes are ({1,3}) and ({2,4}). Step 3: Equivalence classes must form non-overlapping groups.

Step 1: The statement connects two related pairs to a third pair. Step 2: This is exactly transitivity. Step 3: An equivalence relation must be reflexive, symmetric, and transitive.

Step 1: The condition (a-b=0) means (a=b). Step 2: Equality is reflexive, symmetric, and transitive. Step 3: Equality is the standard example of an equivalence relation.

Step 1: Division by (3) gives possible remainders (0,1,2). Step 2: The classes are ({3,6},{1,4},{2,5}). Step 3: In modulo relations, equivalence classes are based on remainders.

Step 1: An equivalence relation is defined by three properties. Step 2: One or two properties are not enough. Step 3: Always test reflexivity, symmetry, and transitivity separately.

Step 1: The universal relation contains all ordered pairs from (A). Step 2: Hence the required pairs for reflexivity, symmetry, and transitivity are present. Step 3: On any set, (A\times A) is an equivalence relation.

Step 1: Reflexivity requires ((1,1),(2,2),(3,3)). Step 2: The empty relation has no pairs, so reflexivity fails. Step 3: On a non-empty set, the empty relation cannot be an equivalence relation.

Step 1: (a+a=2a) is always even, so the relation is reflexive. Step 2: If (a+b) is even, then (b+a) is even, so it is symmetric. Step 3: If (a+b) and (b+c) are even, then (a) and (c) have the same parity, so it is transitive.