If (A={1,2,3,4}) and (R={(a,b)\in A\times A:a+b=5}), then how many elements are in (R)?
Correct answer: C
(R={(1,4),(2,3),(3,2),(4,1)}), so the count is (4). In exams, remember that order matters in ordered pairs.
Mathematics
Relations
संबंध
In this Class 11 Mathematics topic, students learn how a relation connects elements of one set with elements of another, building the foundation for the chapter Relations and Functions. The topic introduces ordered pairs, Cartesian products, and the representation of relations as subsets of a Cartesian product. Students also examine the domain, codomain, and range of a relation and learn to interpret relations through rosters, diagrams, and set notation, preparing them to distinguish relations from functions.
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Expert · Level 3
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- (2)
- (3)
- (4)
- (5)
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- Symmetric relation only
- Equivalence relation
- Transitive relation only
- Antisymmetric but not symmetric relation
Expert · Level 3
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- Reflexive but not symmetric
- Symmetric but not reflexive
- Neither reflexive nor symmetric
- Both reflexive and symmetric
Expert · Level 3
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- Only symmetric
- Reflexive and transitive
- Only asymmetric
- Symmetric and empty
Expert · Level 3
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- Symmetric but not reflexive
- Reflexive but not symmetric
- Reflexive and transitive
- Neither symmetric nor reflexive
Expert · Level 3
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- (R) is reflexive
- (R) is universal
- (R) is an empty relation
- (R) is identity relation
Expert · Level 3
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- Only reflexive
- Equivalence relation
- Only symmetric
- Not transitive
Expert · Level 3
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- \(\{4,5,6\}\)
- \(\{1,2,3\}\)
- \(\{1,2,4,5\}\)
- \(\{2,4\}\)
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- \(\{1,2,3\}\)
- \(\{4,5,6\}\)
- \(\{2,4,6\}\)
- \(\{1,4\}\)
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- (1,4)
- (2,4)
- (3,3)
- (4,2)
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- Reflexive and transitive but not symmetric
- Symmetric and reflexive but not transitive
- Only symmetric
- Neither reflexive nor transitive
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- (2^3)
- (2^6)
- (2^9)
- (3^2)
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- (2^{12})
- (2^{16})
- (2^4)
- (4^2)
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- It is reflexive
- It is symmetric
- It is transitive
- It is universal
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- ({(x,y):x-y=3})
- ({(x,y):y-x=3})
- ({(x,y):x+y=3})
- ({(x,y):xy=3})
Expert · Level 3
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- Reflexive and symmetric
- Symmetric but not reflexive
- Reflexive but not symmetric
- Transitive and reflexive
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- (5)
- (10)
- (15)
- (20)
Expert · Level 3
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- Reflexive and symmetric
- Irreflexive and transitive
- Symmetric and transitive
- Reflexive and universal
Expert · Level 3
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- It is an equivalence relation
- It is only symmetric
- It is not reflexive
- It is not transitive
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- (1, 2)
- (2, 2)
- (3, 4)
- (4, 1)
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- (\varnothing)
- ({(1,1),(2,2),(3,3)})
- (A\times A)
- ({(1,2),(2,3),(3,1)})
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- (2)
- (3)
- (5)
- (6)
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- {1, 2, 3}
- {x, y}
- {x}
- {y}
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- 8
- 9
- 10
- 11
Expert · Level 3
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- ((2,3))
- ((4,6))
- ((1,4))
- ((2,4))
Let R be a relation on a set A defined by: for every a, b ∈ A, aRb if and only if a = b. What type of relation is R?
Correct answer: B
This is the equality relation. Since a=a, it is reflexive. If aRb, then a=b, so b=a and bRa; hence it is symmetric. Also, aRb and bRc imply a=c, so it is transitive. Therefore, R is an equivalence relation. Exam tip: check reflexive, symmetric and transitive properties together.
On set (A={1,2,3}), (R={(1,1),(2,2),(3,3),(1,2)}). Which statement is correct for (R)?
Correct answer: A
All ((a,a)) pairs are present, so (R) is reflexive. Since ((1,2)) is present but ((2,1)) is not, it is not symmetric.
If (A={1,2,3,4}) and (R={(a,b):a\le b}), which property is certainly satisfied by (R)?
Correct answer: B
For every (a\in A), (a\le a), so it is reflexive. If (a\le b) and (b\le c), then (a\le c), so it is transitive.
For the relation (R={(1,2),(2,1),(2,3),(3,2)}) on (A={1,2,3}), choose the correct option.
Correct answer: A
Every reverse pair is present, so (R) is symmetric. But ((1,1),(2,2),(3,3)) are missing, so it is not reflexive.
If (R=\varnothing) on (A={2,4,6}), which statement is correct?
Correct answer: C
There is no ordered pair in (R), so it is an empty relation. On non-empty (A), it is not reflexive.
On (A={1,2,3,4,5}), (R={(a,b):a-b\text{ is divisible by }2}). What type of relation is (R)?
Correct answer: B
Elements with the same parity are related, so reflexive, symmetric, and transitive properties all hold. Hence (R) is an equivalence relation.
If \(R\subseteq A\times B\) and \(R=\{(1,4),(2,4),(2,5),(3,6)\}\), what is the domain of \(R\)?
Correct answer: B
The domain of a relation is the set of all first components of its ordered pairs. Here the first components are 1, 2, 2 and 3; removing the repeated 2 gives \(\{1,2,3\}\). Option A is the range, while options C and D do not correctly include all first components. Exam tip: use the first coordinate for the domain and the second coordinate for the range.
If the relation \(R=\{(1,4),(2,4),(2,5),(3,6)\}\), what is the range of \(R\)?
Correct answer: B
The range of a relation is the set of the second components of its ordered pairs. Here the second components are 4, 4, 5, and 6; since repeated elements are written only once in a set, the range is \(\{4,5,6\}\). Option A is the set of first components, or the domain. Exam tip: in an ordered pair \((a,b)\), select the second component \(b\) to find the range.
On the set \(A=\{1,2,3,4\}\), the relation \(R=\{(a,b):a\mid b\}\) is defined. Which of the following ordered pairs does not belong to \(R\)?
Correct answer: D
An ordered pair \((a,b)\) belongs to \(R\) only when \(a\) divides \(b\), meaning that \(b/a\) is an integer. We have \(1\mid4\), \(2\mid4\), and \(3\mid3\), but \(4\nmid2\); hence, \((4,2)\notin R\). Exam tip: in a divisibility relation, the first component is the divisor and the second component is the dividend.
For (R={(a,b):a\mid b}) on (A={1,2,3,4}), which statement is correct?
Correct answer: A
Every (a\mid a), so it is reflexive, and divisibility is transitive. But (1\mid2) while (2\nmid1), so it is not symmetric.
If (A={1,2,3}), how many relations are possible on (A)?
Correct answer: C
Direct answer: Option C, 2^9. A relation on A is any subset of the Cartesian product A×A. Since A has 3 elements, the ordered pairs are 3×3=9: each first element can be paired with each of the three second elements. For every one of these 9 pairs, we have two independent choices: include it in the relation or leave it out. Therefore the number of subsets, and hence relations, is 2×2×... nine times = 2^9=512. Option A, 2^3, counts choices for only three objects and ignores the ordered-pair construction. Option B, 2^6, uses the wrong number of possible pairs. Option C is correct. Option D, 3^2, reverses the exponent idea and is not the number of subsets. No extra condition such as reflexivity or symmetry is imposed, so all subsets count. Memory cue: an n-element set has n^2 ordered pairs and therefore 2^(n^2) relations.
If (|A|=4), how many relations on (A) must contain ((a,a)) for all (a\in A)?
Correct answer: A
In a reflexive relation, (4) diagonal pairs are fixed. The remaining (16-4=12) pairs are free, so the number is (2^{12}).
On (A={1,2,3,4}), (R={(a,b):|a-b|=1}). Which statement about (R) is correct?
Correct answer: B
If (|a-b|=1), then (|b-a|=1), so it is symmetric. But ((1,2)) and ((2,3)) hold while ((1,3)) does not, so it is not transitive.
If (R={(x,y)\in \mathbb{Z}\times\mathbb{Z}:x-y=3}), how can (R^{-1}) be written?
Correct answer: B
In the inverse, order is reversed, so the original condition (a-b=3) becomes (y-x=3). Do not forget the direction when variables are swapped.
For the relation (R={(a,b):a+b\text{ is odd}}) on (A={1,2,3,4}), which statement is correct?
Correct answer: B
If (a+b) is odd, then (b+a) is also odd, so it is symmetric. But (a+a=2a) is even, so it is not reflexive.
If (A={1,2,3,4,5}) and (R={(a,b):a<b}), how many ordered pairs are in (R)?
Correct answer: B
For (5) elements, the number of pairs with (a<b) is (\binom{5}{2}=10). In such questions, count each unordered pair in the correct ordered direction.
Choose the correct option for (R={(a,b):a<b}) on (A={1,2,3,4,5}).
Correct answer: B
(a<a) is never true, so it is irreflexive. If (a<b) and (b<c), then (a<c), so it is transitive.
If (R={(x,y)\in \mathbb{R}\times\mathbb{R}:x^2=y^2}), which statement is correct for (R)?
Correct answer: A
(x^2=x^2), (x^2=y^2\Rightarrow y^2=x^2), and equality is transitive. Hence it is an equivalence relation.
Which of the following ordered pairs belongs to the identity relation I_A defined on A = {1, 2, 3, 4}?
Correct answer: B
The identity relation on A is I_A = {(a, a) : a ∈ A}; hence, both components of every ordered pair must be equal. Therefore, (2, 2) belongs to I_A. The pairs (1, 2), (3, 4), and (4, 1) are not identity pairs because their components differ. Exam tip: In an identity relation, every element is related only to itself.
If (A={1,2,3}), what is the smallest reflexive relation on (A)?
Correct answer: B
For reflexivity, all diagonal pairs are compulsory. The smallest relation keeps only those required pairs.
If (R) is a relation with (\operatorname{Dom}(R)={2,3,5}) and (\operatorname{Ran}(R)={7,11}), what is the minimum possible number of pairs in (R)?
Correct answer: B
All three domain elements must appear in some pair, so at least (3) pairs are needed. The (2) range elements can be covered within these (3) pairs.
If A = {1, 2, 3}, B = {x, y}, and R = {(1, x), (2, x), (3, y)}, what is the range of R?
Correct answer: B
The range of a relation is the set of all second components of its ordered pairs. The second components in R are x, x, and y; removing the repetition gives {x, y}. The set {1, 2, 3} is the domain, while {x} and {y} omit one of the attained values. Exam tip: To find the range, collect the second element from every ordered pair and write repeated elements only once.
Let \(A=\{1,2,3,4\}\) and define the relation \(R=\{(a,b)\in A\times A:a+b\le 5\}\). How many ordered pairs does \(R\) contain?
Correct answer: C
For each value of \(a\), count the values of \(b\) satisfying \(a+b\le 5\). When \(a=1\), there are 4 choices for \(b\); for \(a=2,3,4\), there are 3, 2, and 1 choices, respectively. Thus, \(|R|=4+3+2+1=10\), so option C is correct. Exam tip: For a relation defined by an inequality, count the admissible second components for each possible first component and then add them.
If (A={1,2,3,4,6}) and (R={(a,b):\operatorname{lcm}(a,b)=6}), which pair belongs to (R)?
Correct answer: A
(\operatorname{lcm}(2,3)=6), so ((2,3)\in R). In the other options, the (\operatorname{lcm}) is not (6).
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