Equivalence relation

Step 1: Every element is in the same part as itself, so the relation is reflexive. Step 2: If two elements are in the same part, reversing the order keeps them in the same part. Step 3: A relation formed by partitions is usually an equivalence relation.

Step 1: An equivalence relation groups elements with the same property. Step 2: Thus, the set is divided into distinct equivalence classes. Step 3: Thinking in terms of classes helps solve these questions faster.

Step 1: The equivalence class of (a) is the set of all elements related to (a). Step 2: In an equivalence relation, this class forms a clear separate block. Step 3: When asked for a class, first understand the condition of the relation.

Step 1: Every integer is congruent to itself because (a-a=0). Step 2: Having the same remainder is preserved when order is reversed and through a chain. Step 3: Same-remainder modulo relations are standard examples of equivalence relations.

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: If two numbers have the same parity as a middle number, they have the same parity with each other.

Step 1: The pair ((1,2)) is present in the relation. Step 2: For symmetry, the reverse pair ((2,1)) must also be present, but it is missing. Step 3: To test symmetry, check the reverse of every non-identical pair.

Step 1: Identity pairs like ((a,a)) are connected with the reflexive property. Step 2: Such a pair must exist for every element. Step 3: Reflexivity does not need comparison with another element.

Step 1: In the symmetric property, the relation remains true when the order is reversed. Step 2: So ((a,b)) must come with ((b,a)). Step 3: Remember it as the reverse-pair rule.

Step 1: Transitivity forms a chain of two related pairs. Step 2: From ((a,b)) and ((b,c)), the direct pair ((a,c)) is required. Step 3: Identify it by matching the middle element (b).

Step 1: On division by (2), (1) and (3) have the same remainder. Step 2: So the class of (1) contains (1) and (3). Step 3: To find an equivalence class, look for elements with the same remainder as the given element.

Step 1: On division by (3), (2) leaves remainder (2). Step 2: (5) also leaves remainder (2), so the class of (2) is ({2,5}). Step 3: In modulo questions, making a remainder table is an easy method.

Step 1: A person is born in the same year as themselves. Step 2: If one person has the same birth year as another, the reverse is also true, and the chain condition also holds. Step 3: Conditions based on sameness usually indicate equivalence relations.

Step 1: (a>a) is never true, so greater than is not reflexive. Step 2: If (a>b), then (b>a) is not true, so it is not symmetric. Step 3: For comparison relations, check all equivalence properties carefully.

Step 1: Every natural number is equal to itself, so the relation is reflexive. Step 2: Equality remains true when the order is reversed. Step 3: The equality relation is always a safe example of an equivalence relation.

Step 1: (|a|=|a|) is always true, so it is reflexive. Step 2: If (|a|=|b|), then (|b|=|a|) is also true. Step 3: Equality of values is also transitive, so this is an equivalence relation.

Step 1: Since (a^2=a^2), the relation is reflexive. Step 2: Equality can be reversed and passed through a chain. Step 3: For a condition like (a^2=b^2), test the three properties through equality.

Step 1: The possible remainders on division by (3) are (0,1,2). Step 2: Each remainder forms a separate class. Step 3: In modulo (n), we usually get (n) remainder classes if all remainders occur.

Step 1: An equivalence relation divides a set into clear blocks. Step 2: So two classes are either the same or have no common element. Step 3: Think of equivalence classes as parts of a partition.

Step 1: Reflexivity is a compulsory condition for an equivalence relation. Step 2: Therefore, every equivalence relation must be reflexive. Step 3: For direct questions like this, remembering the definition is most useful.

Step 1: Symmetry is a required part of an equivalence relation. Step 2: So every related ordered pair must have its reverse pair in the relation. Step 3: Forgetting reverse pairs is a common mistake in equivalence relation questions.