Equivalence relation

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

Step 1: Symmetry is also necessary for an equivalence relation. Step 2: Without symmetry, all three properties are not satisfied. Step 3: Even if two properties hold, always check the third one.

Step 1: Reflexivity is a compulsory condition for an equivalence relation. Step 2: Since reflexivity is missing, the relation cannot be an equivalence relation. Step 3: It is important to remember all three defining properties.

Step 1: (2) is even and divisible by (2). Step 2: In the given set, (2,4,6) are all divisible by (2). Step 3: While forming a class, include all elements with the same property.

Step 1: (3) is odd and leaves remainder (1) on division by (2). Step 2: (1,3,5) all leave the same remainder. Step 3: Separating even and odd classes is an easy method.

Step 1: An equivalence class is a group of elements related to one another. Step 2: Therefore, two elements in the same class form a pair in the relation. Step 3: For elements inside the same class, write the related pair.

Step 1: Elements from different equivalence classes are not related. Step 2: Therefore, ((a,b)) will not belong to (R). Step 3: If the classes are different, treat the pair as outside the relation.

Step 1: The set (A) has (3) elements. Step 2: (A\times A) has (3\times 3=9) ordered pairs. Step 3: In a universal relation, count all possible ordered pairs.

Step 1: Reflexivity is connected with identity pairs. Step 2: For every element (a), ((a,a)\in R) must be present. Step 3: To check reflexivity, list all identity pairs.

Step 1: In symmetry, the relation remains true after reversing the order. Step 2: So ((a,b)) must come with ((b,a)). Step 3: Identify symmetry by checking the reverse pair.

Step 1: Transitivity forms a link between two pairs. Step 2: From ((a,b)) and ((b,c)), ((a,c)) must follow. Step 3: Identify the common middle element before answering.

Step 1: ((2,1)\in R) and ((1,3)\in R). Step 2: Transitivity requires ((2,3)\in R), but it is missing. Step 3: In transitivity, look for the missing direct pair.

Step 1: On division by (2), there are only two remainders, (0) and (1). Step 2: Remainder (0) gives even numbers, and remainder (1) gives odd numbers. Step 3: Remember modulo (2) through even and odd classes.

Step 1: (0) leaves remainder (0) on division by (3). Step 2: In the given set, (3) also leaves remainder (0). Step 3: While finding the class, choose elements with the same remainder.

Step 1: In a partition, every element belongs to some part. Step 2: Being in the same part is reflexive, symmetric, and transitive. Step 3: A relation formed from a partition is a standard form of an 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.