Equivalence relation

Step 1: Reflexivity needs (aRa) for every (a). Step 2: (a+a=2a) is always even, not odd. Step 3: Once reflexivity fails, the relation cannot be an equivalence relation.

Step 1: Elements related to (2) must have absolute value (2). Step 2: Both (-2) and (2) have absolute value (2). Step 3: To find an equivalence class, apply the relation condition to the chosen element.

Step 1: Every point has the same (x)-coordinate as itself. Step 2: If two points have the same (x)-coordinate, the order does not matter. Step 3: Equality of (x)-coordinates also passes through a third point, so transitivity holds.

Step 1: An equivalence class is the block that contains the given element. Step 2: (4) lies in ({1,4}). Step 3: When a partition is given, choose the block containing the element.

Step 1: A partition has non-empty, non-overlapping blocks whose union is the whole set. Step 2: ({1,2}) and ({3,4}) are disjoint and cover (A). Step 3: Every equivalence relation creates a partition.

Step 1: In an equivalence relation, related elements belong to the same class. Step 2: Therefore (aRb) implies ([a]=[b]). Step 3: If two equivalence classes overlap, they are actually equal.

Step 1: (1) and (3) are related to each other, so they form one class. Step 2: (2) and (4) are related only to themselves. Step 3: Connected related elements form groups, while isolated elements form singleton classes.

Step 1: Every number has the same last digit as itself. Step 2: Having the same last digit is symmetric. Step 3: If two numbers share the same last digit through a middle number, the first and third also share it.

Step 1: The last digit of (23) is (3). Step 2: So numbers like (3,13,23,33) belong to the same class. Step 3: The class is built using the defining property of the relation.

Step 1: The identity relation contains all pairs ((a,a)). Step 2: Since only identical elements are paired, symmetry and transitivity also hold. Step 3: The identity relation is an equivalence relation on any set.

Step 1: Division by (5) gives only remainders (0,1,2,3,4). Step 2: Integers with the same remainder form one class. Step 3: Although integers are infinite, there are only (5) classes.

Step 1: Symmetry is necessary for an equivalence relation. Step 2: (2\leq 3) is true, but (3\leq 2) is false. Step 3: Order relations often fail symmetry.

Step 1: For (|1-b|) to be even, (b) must have the same parity as (1). Step 2: (1,3,5) are odd, so they are related to (1). Step 3: An even difference relation groups numbers by parity.

Step 1: An equivalence relation is reflexive. Step 2: Therefore (aRa), so (a\in [a]). Step 3: Equivalence classes are never empty.

Step 1: Every student is in the same class as himself or herself. Step 2: If one student is in the same class as another, the reverse is also true. Step 3: The same-class relation is transitive, so it is an equivalence relation.

Step 1: The pairs ((2,1)) and ((1,3)) are present. Step 2: Transitivity would require ((2,3)). Step 3: In transitivity checks, identify the middle element and demand the connecting pair.

Step 1: (a^2=a^2), so the relation is reflexive. Step 2: If (a^2=b^2), then (b^2=a^2), so it is symmetric. Step 3: If (a^2=b^2) and (b^2=c^2), then (a^2=c^2), so it is transitive.

Step 1: The square of (-3) is (9). Step 2: The integers whose square is (9) are (-3) and (3). Step 3: Equal-square classes usually contain both a number and its negative.

Step 1: Same remainder modulo (2) means same parity. Step 2: (2) is even and (5) is odd, so their remainders are different. Step 3: In such options, compare remainders first.

Step 1: An equivalence relation divides a set into classes. Step 2: If two classes share an element, they become equal; otherwise they are disjoint. Step 3: This idea is central in partition-based questions.