Equivalence relation

Step 1: (a\le a) makes the relation reflexive. Step 2: But (2\le 3) is true while (3\le 2) is false, so symmetry fails. Step 3: Order relations often fail symmetry.

Step 1: Reflexivity would require (a<a). Step 2: No number is less than itself, so reflexivity fails. Step 3: Strict inequality relations are not equivalence relations.

Step 1: Real numbers include negative numbers and zero. Step 2: If (a) is negative, (aRa) is false because (a) is not positive. Step 3: Reflexivity must hold for every element of the set.

Step 1: Every number has the same last digit as itself. Step 2: Same last digit is symmetric and transitive. Step 3: Think of last digit as remainder classes modulo (10).

Step 1: The definition of an equivalence relation is based on three properties. Step 2: These are reflexive, symmetric, and transitive. Step 3: For definition questions, remember all three together.

Step 1: Same parity gives one class of odd numbers and one class of even numbers. Step 2: In (A), the classes are {(1,3)} and {(2,4)}. Step 3: To count classes, first form the groups clearly.

Step 1: Every integer has the same absolute value as itself, so the relation is reflexive. Step 2: Equality of absolute values is symmetric and transitive. Step 3: Relations based on equality usually become equivalence relations after all three checks.

Step 1: The relation holds only when both elements are in the same class. Step 2: (3) and (5) both lie in ({3,5}), so ((3,5)\in R). Step 3: In class-based questions, first locate the two elements in the given groups.

Step 1: All self-pairs are present and reverse pairs are also visible. Step 2: But ((1,2)) and ((2,3)) should imply ((1,3)), which is missing. Step 3: For transitivity, look for pairs whose middle element matches.

Step 1: (4) leaves remainder (0) when divided by (4). Step 2: In the given set, (8) also leaves remainder (0), so (4) and (8) are in the same class. Step 3: In remainder-class questions, use only elements from the given set.

Step 1: Every element is related to itself, so the relation is reflexive. Step 2: Since only identical ordered pairs are present, symmetry and transitivity also hold. Step 3: In exams, always check all three properties for an equivalence relation.

Step 1: An equivalence relation is identified by three properties. Step 2: These are reflexive, symmetric, and transitive. Step 3: If even one property fails, the relation is not an equivalence relation.

Step 1: The relation has all identity pairs, so it is reflexive. Step 2: The pair ((1,2)) has ((2,1)), and the small block ({1,2}) is closed under transitivity. Step 3: Check the pair block carefully before rejecting the relation.

Step 1: Since (a-a=0) is even, the relation is reflexive. Step 2: If (a-b) is even, then (b-a) is also even, and the sum of two even differences is even. Step 3: Relations based on parity often form equivalence relations.

Step 1: Every number belongs to the same parity class as itself, so the relation is reflexive. Step 2: If (a) and (b) are in the same class, then (b) and (a) are also in the same class. Step 3: Class-based relations usually make transitivity easy to verify.

Step 1: (A\times A) contains every possible ordered pair. Step 2: So all identity pairs, reverse pairs, and required transitive pairs are present. Step 3: Remember the universal relation as a common example of an equivalence relation.

Step 1: In the equality relation, every element is related to itself. Step 2: If (a=b), then (b=a), and if (a=b) and (b=c), then (a=c). Step 3: The equality relation is the simplest example of an equivalence relation.

Step 1: The symmetric property says that if ((a,b)\in R), then ((b,a)\in R). Step 2: Every equivalence relation must be symmetric. Step 3: Looking for the reverse pair is the easiest way to test symmetry.

Step 1: In transitivity, the second element of the first pair matches the first element of the second pair. Step 2: Therefore, ((a,b)) and ((b,c)) imply ((a,c)). Step 3: In transitivity questions, focus on the middle element.

Step 1: For reflexivity, every element (a) of (A) must have ((a,a)\in R). Step 2: Here, ((3,3)) is missing for the element (3). Step 3: When checking equivalence, first check all identity pairs.