Relations

Correct answer: C. ( (3,4) )

Since (3) does not divide (4), ( (3,4) \notin R ). In exams, check the direction of divisibility carefully.

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

The direct answer is option A, {(3,1),(4,2),(5,3)}. The inverse relation is obtained by reversing the order of every ordered pair: (a,b) becomes (b,a). Start with (1,3), which becomes (3,1). Next, (2,4) becomes (4,2). Finally, (3,5) becomes (5,3). Collecting these gives exactly option A. Option B is the original relation, so no components have been reversed. Option C contains incorrect changes: it does not reverse the given pairs in a consistent way. Option D contains the original pairs in reverse listing order, but merely changing the order in which pairs are written is not the same as taking the inverse; each pair itself must be reversed. Therefore A is correct. Exam cue: inverse means flip every arrow, not merely reorder the list.

Correct answer: B. Because ( (3,3)\notin R )

For a reflexive relation, ( (a,a)\in R ) is required for every (a\in A). Here ( (3,3)\notin R ), so it is not reflexive.

Correct answer: A. It is symmetric

For every ( (a,b)\in R ), ( (b,a)\in R ) is also present, so (R) is symmetric. In exams, match each pair with its reverse.

Correct answer: C. It satisfies the transitive condition for the given pairs

Since ( (1,2)\in R ) and ( (2,3)\in R ) imply ( (1,3)\in R ). For transitivity, connect the middle element carefully.

Correct answer: A. Same parity

If (a-b) is even, then (a) and (b) have the same parity. In exams, make odd and even groups separately.

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

A relation from (A) to (B) must be a subset of (A\times B). So first components must come from (A), and second components must come from (B).

Correct answer: B. (2^{12}-1)

Total relations are (2^{|A\times B|}=2^{12}), and removing the empty relation gives (2^{12}-1). In exams, remember to subtract (1) for non-empty relations.

Correct answer: B. (b=2a)

In every ordered pair, the second component is (2) times the first, so (b=2a). In exams, test the rule on all pairs.

Correct answer: A. ( \varnothing )

An empty relation has no ordered pair, so it is ( \varnothing ). In exams, keep empty relation and identity relation separate.

Correct answer: A. ( (2,1) )

Direct answer: option A, (2,1), belongs to R. The defining condition is a=b+1, so substitute the first component for a and the second for b. A: for (2,1), 2=1+1, which is true; hence it belongs. B: for (1,2), the condition would require 1=2+1=3, false. C: for (3,5), it would require 3=5+1=6, false. D: for (4,4), it would require 4=4+1=5, false. All numbers shown are natural numbers, so the membership test is simply the equation and the order of the components. Do not use b=a+1, because that reverses the given relation and changes the answer. The relation contains pairs in which the first entry is exactly one greater than the second. Memory cue: “first = second + 1.”

Correct answer: A. Yes, because ( (1,1),(2,2),(3,3)\in R )

For reflexivity, all pairs ( (a,a) ) are required, and they are present here. The extra pair ( (1,2) ) does not break reflexivity.

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

For symmetry, the reverse of ( (1,2) ) is ( (2,1) ), and the reverse of ( (2,2) ) is itself. ( (1,1) ) is not necessary.

Correct answer: C. (10)

For (a\leq b), the number of pairs is (4+3+2+1=10). In exams, include diagonal pairs ( (a,a) ) also.

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

The first component must be greater than some smaller (b), so (2,3,4) appear. (1) is not greater than any (b\in A).

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

The second component must have some greater (a\in A), so (1,2,3) appear. There is no element in (A) greater than (4).

Correct answer: A. When (R\subseteq A\times B)

A relation from (A) to (B) is any subset of (A\times B). In exams, treat a relation as a set of ordered pairs.

Correct answer: B. Reflexive and symmetric relation

It is reflexive because ( (1,1),(2,2),(3,3) ) are present, and symmetric because ( (1,2),(2,1) ) are paired. It is not universal because all pairs are not present.

Correct answer: A. ( (1,3) )

The direct answer is option A, (1,3). Transitivity requires that if (a,b) and (b,c) belong to R, then (a,c) must also belong to R. Here (1,2) is present and (2,3) is present. The middle entries match at 2, so the required shortcut is from 1 directly to 3: (1,3). Adding it makes the displayed chain satisfy the transitive condition. Option A is correct. Option B, (3,1), goes backward and is not demanded by the rule. Option C, (2,1), reverses the first relation and is irrelevant. Option D, (3,2), reverses the second relation and is also irrelevant. Notice that transitivity does not mean adding every possible pair; it means adding the pair forced by a two-step chain. Since the set has only these three elements and the relation has the chain 1 to 2 to 3, (1,3) is the needed addition. Remember: aRb and bRc imply aRc.

Correct answer: C. (6)

The pairs are ( (1,2),(2,1),(2,3),(3,2),(3,4),(4,3) ), so the count is (6). In exams, take both directions because of absolute difference.

Correct answer: B. ( (4,2) )

(4) is a multiple of (2), so ( (4,2)\in R ). In exams, understand the direction of multiple of and divides separately.

Correct answer: A. Domain ( {1,2,3} ), range ( {4,5,6} )

The domain is formed from first components and the range from second components. Therefore domain is ( {1,2,3} ) and range is ( {4,5,6} ).

Correct answer: D. ( (1,3) )

(1+3=4) is even, so ( (1,3)\notin R ). In exams, an odd sum needs one odd and one even element.

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

The smallest reflexive relation contains only all pairs ( (a,a) ). It is also called the identity relation.

Correct answer: C. ( A\times A )

The largest relation is (A\times A) because it contains all possible ordered pairs. It is called the universal relation.