Equivalence relation

Correct answer: A. ({..., -4,2,8,14,20,26,...})

Step 1: Integers related to (20) must differ from (20) by a multiple of (6). Step 2: (20) gives remainder (2) on division by (6), so all integers with remainder (2) are in its class. Step 3: In modulo-class questions, first find the remainder of the given element.

Correct answer: D. ((1,5))

Step 1: Only pairs with the same remainder belong to this relation. Step 2: (1) has remainder (1), while (5) has remainder (2), so ((1,5)) is not in the relation. Step 3: For each option, compare the remainders after division by (3).

Correct answer: C. (8)

Step 1: In an equivalence relation, every element of a class is related to every element of the same class. Step 2: ({a,c}) contributes (2^2=4) pairs and ({b,d}) contributes (2^2=4) pairs. Step 3: The total is (4+4=8), with no cross-class pairs.

Correct answer: A. ([x]=[y])

Step 1: A key property of equivalence classes is that they are either disjoint or exactly equal. Step 2: ([x]\cap[y]\neq\varnothing) means the two classes share at least one element. Step 3: Therefore the two classes must be equal; this is a useful rule in partition questions.

Correct answer: A. ({...,1,8,15,22,29,...})

Step 1: Integers related to (15) must differ from (15) by a multiple of (7). Step 2: (15) gives remainder (1) on division by (7), so all integers with remainder (1) are in the same class. Step 3: First identify the remainder to find the equivalence class quickly.

Correct answer: A. ({1,4,7})

Step 1: (4) gives remainder (1) on division by (3). Step 2: (1,4,7) all have remainder (1), so they form the same class. Step 3: In a finite set, include only elements that belong to the given set.

Correct answer: A. ({1,4},{2},{3})

Step 1: (1) and (4) are related to each other, so they form one class. Step 2: (2) and (3) are related only to themselves, so they form singleton classes. Step 3: Group mutually related elements together.

Correct answer: C. (10)

Step 1: Within each equivalence class, every element is related to every element. Step 2: ({p,q,r}) contributes (3^2=9) pairs and ({s}) contributes (1^2=1) pair. Step 3: The total number of pairs is (9+1=10).

Correct answer: C. (8)

Step 1: Division by (8) gives remainders (0,1,2,3,4,5,6,7). Step 2: Integers with the same remainder lie in the same equivalence class. Step 3: Although there are infinitely many integers, there are (8) distinct classes.

Correct answer: A. Equivalence relation

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

Correct answer: A. ({-1,1})

Step 1: The square of (1) is (1). Step 2: In the given set, both (-1) and (1) have square (1). Step 3: In square-equality relations, a number and its negative often belong to the same class.

Correct answer: A. ((1,4))

Step 1: (|1-4|=3), which is divisible by (3). Step 2: Therefore ((1,4)) belongs to the relation. Step 3: Find the absolute difference and then test divisibility.

Correct answer: A. They are disjoint

Step 1: Equivalence classes are either equal or disjoint. Step 2: If (a) and (b) are not related, they cannot be in the same class. Step 3: Hence their classes are disjoint.

Correct answer: B. ({1,3,5},{2,4})

Step 1: A partition has non-empty blocks that do not overlap. Step 2: ({1,3,5}) and ({2,4}) are disjoint and cover the whole set. Step 3: Equivalence classes always form a valid partition.

Correct answer: A. Because ((2,3)) is missing

Step 1: The pairs ((2,1)) and ((1,3)) are in the relation. Step 2: Transitivity requires ((2,3)), but it is missing. Step 3: While checking equivalence, use two connected pairs to identify the required third pair.

Correct answer: A. Equivalence relation

Step 1: (a-a=0) is an integer, so the relation is reflexive. Step 2: If (a-b) is an integer, then (b-a=-(a-b)) is also an integer. Step 3: The sum of two integer differences is also an integer, so transitivity holds.

Correct answer: A. (\frac{5}{2})

Step 1: (\frac{5}{2}-\frac{1}{2}=2). Step 2: Since (2) is an integer, (\frac{5}{2}) and (\frac{1}{2}) are in the same equivalence class. Step 3: In such questions, subtract the two numbers.

Correct answer: A. ({2,4,6})

Step 1: For (a+b) to be even, the two numbers must have the same parity. Step 2: Since (2) is even, the elements related to (2) are (2,4,6). Step 3: An even-sum relation creates even and odd classes.

Correct answer: A. A partition of set (A)

Step 1: Every element belongs to some equivalence class. Step 2: Distinct equivalence classes are disjoint and together cover (A). Step 3: Hence the collection of all equivalence classes is a partition of (A).

Correct answer: D. ((2,3))

Step 1: The relation from a partition contains pairs only within the same block. Step 2: (2) and (3) are in different blocks. Step 3: Cross-block pairs are not included, so ((2,3)) is not in the relation.

Correct answer: B. ({1},{2,3,4})

Step 1: Equivalence classes must cover the whole set. Step 2: ({1}) and ({2,3,4}) are disjoint and cover all of (A). Step 3: Blocks that overlap or miss elements cannot be equivalence classes.

Correct answer: A. Equivalence relation

Step 1: Every element is in the same group as itself, so reflexivity holds. Step 2: Being in the same group is symmetric. Step 3: The two classes are ({3,6}) and ({1,2,4,5}), so the relation is an equivalence relation.

Correct answer: A. ({1,2,4,5})

Step 1: (5) is not a multiple of (3). Step 2: Therefore its class contains all elements that are not multiples of (3). Step 3: To find a class, first identify the group of the given element.

Correct answer: B. Symmetry

Step 1: The statement says that the reverse of every related pair is also related. Step 2: This is the definition of symmetry. Step 3: An equivalence relation also needs reflexivity and transitivity.

Correct answer: A. On integers, (aRb) when (a-b) is even

Step 1: (a-b) being even means the two integers have the same parity. Step 2: This relation is reflexive, symmetric, and transitive. Step 3: Order-type relations often fail symmetry.