Equivalence relation

Step 1: In the identity relation, every element is related only to itself. Step 2: Hence each element forms a separate singleton class. Step 3: The number of classes in an identity relation equals the number of elements.

Step 1: Every integer is either positive or not positive, so it is related to itself within its group. Step 2: Being in the same group is symmetric. Step 3: It creates two classes: positive integers and non-positive integers.

Step 1: (0) is not positive. Step 2: Therefore (0) is related to all non-positive integers. Step 3: Read the grouping condition carefully to identify the class.

Step 1: Every object has the same colour as itself. Step 2: If one object has the same colour as another, the reverse is also true. Step 3: Having the same colour is transitive, so it is an equivalence relation.

Step 1: (\lfloor a\rfloor=\lfloor a\rfloor), so reflexivity holds. Step 2: Equality of the greatest integer value is symmetric. Step 3: If two values are equal to the same middle value, they are equal to each other, so transitivity holds.

Step 1: (\lfloor 2.7\rfloor=2). Step 2: Real numbers with greatest integer value (2) lie in ([2,3)). Step 3: For floor-based classes, the left endpoint is included and the right endpoint is excluded.

Step 1: In the given set, (2,3,5) are prime. Step 2: (4) is not prime, so its class is ({1,4,6}). Step 3: For classification relations, first separate the groups clearly.

Step 1: In an equivalence relation, only elements within the same class are related. Step 2: (2) is in ({1,2}), while (5) is in ({3,4,5}). Step 3: Elements from different classes do not form related ordered pairs.

Step 1: The degree of (x^2+1) is (2). Step 2: All non-zero polynomials of degree (2) are related to it. Step 3: To find an equivalence class, identify the key property of the chosen element.

Step 1: The first component of ((1,2)) is (1). Step 2: In the given set, the ordered pairs with first component (1) are ((1,1)) and ((1,2)). Step 3: In component-based relations, keep the specified component the same to form the class.

Step 1: Every integer related to (25) must differ from (25) by a multiple of (9). Step 2: (25) gives remainder (7) on division by (9), so integers with the same remainder (7) form its class. Step 3: In modulo questions, first find the remainder.

Step 1: (7) gives remainder (3) on division by (4). Step 2: In the given set, only (3) and (7) give remainder (3). Step 3: For a finite set, include only elements from that set.

Step 1: Dividing (4) by (3) gives quotient (1). Step 2: The numbers (3,4,5) all have quotient (1). Step 3: Read carefully whether the relation is based on quotient or remainder.

Step 1: (1,3,5) are mutually related, so they form one class. Step 2: (2) and (4) are related only to themselves, so they form singleton classes. Step 3: Identify connected groups from the ordered pairs.

Step 1: The pairs ((1,2)) and ((2,3)) are in the relation. Step 2: Transitivity requires ((1,3)). Step 3: Identify the middle element and check the pair from the first to the third element.

Step 1: Reflexivity requires every element of the set to be related to itself. Step 2: (3) is in the set, but ((3,3)) is missing. Step 3: If even one diagonal pair is absent, the relation cannot be an equivalence relation.

Step 1: The equivalence class of an element is the block containing that element. Step 2: (3) lies in ({1,3}). Step 3: When a partition is given, directly choose the block containing the element.

Step 1: A class of size (m) contributes (m^2) ordered pairs. Step 2: The total is (4^2+2^2=16+4). Step 3: No pairs are added between different classes, so the answer is (20).

Step 1: A partition must have non-empty, non-overlapping blocks whose union is the whole set. Step 2: The first option has disjoint blocks covering all of (A). Step 3: Equivalence classes always form such a partition.

Step 1: In a partition, different blocks must not share an element. Step 2: In the first option, (2) appears in both ({1,2}) and ({2,3}). Step 3: Equivalence classes do not overlap.