Equivalence relation

Step 1: (2) is even. Step 2: In the set, (4) is the other even number, so it is related to (2). Step 3: Write only the elements related to the chosen element.

Step 1: Every object has the same color as itself. Step 2: Same color is symmetric and transitive. Step 3: A relation based on an identical attribute often gives an equivalence relation.

Step 1: The pair ((1,2)) is in the relation. Step 2: For symmetry, ((2,1)) should also be present, but it is missing. Step 3: One missing reverse pair is enough to fail symmetry.

Step 1: All self-pairs are present, so reflexivity is fine. Step 2: ((1,2)) is present but ((2,1)) is missing, so symmetry fails. Step 3: For equivalence, every non-self pair needs its reverse pair.

Step 1: In the identity relation, each element is related only to itself. Step 2: So its pairs have the form ((a,a)). Step 3: The identity relation is a basic example of an equivalence relation.

Step 1: In the universal relation, every element is related to every element. Step 2: Hence it equals (A\times A). Step 3: The universal relation easily satisfies all three equivalence properties.

Step 1: An equivalence relation must be reflexive. Step 2: Reflexivity gives ((a,a)\in R) for every (a). Step 3: Use the word reflexivity whenever self-pairs are required.

Step 1: The reverse of ((4,7)) is ((7,4)). Step 2: An equivalence relation is symmetric, so the reverse pair must be present. Step 3: Reverse-pair questions usually use symmetry.

Step 1: In ((1,2)) and ((2,5)), the middle element (2) matches. Step 2: By transitivity, ((1,5)) must be present. Step 3: Take the first and last elements to form the transitive pair.

Step 1: Modulo (2), (0) has remainder (0). Step 2: Integers with remainder (0) are even integers. Step 3: So ([0]) is the class of even integers.

Step 1: (1) has remainder (1) when divided by (2). Step 2: Integers with this remainder are odd. Step 3: Modulo (2) gives two classes: even and odd.

Step 1: (1) leaves remainder (1) on division by (3). Step 2: (4) also leaves remainder (1). Step 3: Elements with the same remainder belong to the same equivalence class.

Step 1: (2) leaves remainder (2) when divided by (3). Step 2: (5) also leaves remainder (2). Step 3: While forming the class, use only elements from the given set.

Step 1: (3) leaves remainder (0) when divided by (3). Step 2: (6) also leaves remainder (0). Step 3: Put all elements with remainder (0) in one class.

Step 1: All self-pairs are present, so the relation is reflexive. Step 2: The non-self pairs occur with their reverse pairs. Step 3: It forms the classes {(1,3)} and {(2,4)}, so it is an equivalence relation.

Step 1: Equivalence classes together cover the original set. Step 2: Combining {(1,2)} and {(3)} gives {(1,2,3)}. Step 3: The union of all classes is the original set.

Step 1: An equivalence relation divides a set into separate groups. Step 2: Two distinct classes have no common element. Step 3: If two classes share even one element, they are the same class.

Step 1: In the identity relation, an element is related only to itself. Step 2: Therefore, the class of (a) contains only (a). Step 3: Identity relation gives singleton equivalence classes.

Step 1: In the universal relation, every element is related to every element. Step 2: So the class of any (a) is the whole set. Step 3: Universal relation gives one large equivalence class.

Step 1: Equivalence classes are either equal or disjoint. Step 2: If ([a]\neq[b]), they have no common element. Step 3: Distinct equivalence classes have empty intersection.