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SubjectsMathematics
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.
Correct answer: A. Irreflexive and transitive
Explanation: Since (a-a=0), (aRa) is never true, so it is irreflexive. If (a>b) and (b>c), then (a>c), so it is transitive.
Correct answer: A. (a\leq b)
Explanation: The function (x^3) is strictly increasing, so (a^3\leq b^3) is exactly equivalent to (a\leq b). Use monotonicity to identify relation properties quickly.
Correct answer: A. Symmetric but not reflexive
Explanation: Addition is commutative, so the relation is symmetric. But (a+a=2a) is always even, so (aRa) never holds.
Correct answer: A. Symmetric but not transitive
Explanation: If (a\neq b), then (b\neq a), so it is symmetric. But (1R2) and (2R1) hold while (1R1) does not, so it is not transitive.
Correct answer: A. (a\leq b)
Explanation: The given pairs are exactly those where the first element is less than or equal to the second. Thus it is the usual order (\leq) on (A).
Correct answer: A. ((1,4))
Explanation: By transitivity, ((1,2)) and ((2,4)) force ((1,4)). A closure adds the minimum necessary pairs.
Correct answer: A. (3)
Explanation: The pairs to add are ((1,3),(2,4),(1,4)). Note that newly added ((1,3)) with ((3,4)) also requires ((1,4)).
Correct answer: A. ((1,1),(2,2),(3,3))
Explanation: In a reflexive closure, ((a,a)) must be present for every (a\in A). Therefore all three diagonal pairs are added.
Correct answer: A. ((2,1),(3,2))
Explanation: In a symmetric closure, ((b,a)) is added for every ((a,b)). Thus ((2,1)) is needed for ((1,2)), and ((3,2)) for ((2,3)).
Correct answer: A. Yes, because no distinct reverse pair appears together
Explanation: Antisymmetry is not affected by diagonal pairs. It fails only when ((a,b)) and ((b,a)) both occur for (a\neq b).
Correct answer: A. Equivalence relation
Explanation: Having the same greatest prime factor behaves like equality, so it is reflexive, symmetric, and transitive. Such same invariant conditions often give equivalence relations.
Correct answer: A. ([a]={a+q:q\in\mathbb{Q}})
Explanation: (xRa) means (x-a\in\mathbb{Q}), so (x=a+q) where (q\in\mathbb{Q}). When writing a class, solve the relation condition for the variable.
Correct answer: A. Symmetric but not reflexive
Explanation: Since (a+b=b+a), the relation is symmetric. But ((4,4)\notin R) because (4+4\leq6) is false, so it is not reflexive.
Correct answer: A. (R\cap S) is also an equivalence relation
Explanation: The intersection retains common diagonal pairs, and symmetry and transitivity are also preserved. Hence the intersection of equivalence relations is again an equivalence relation.
Correct answer: A. ({1,3}) and ({2,4})
Explanation: (a+b) is even exactly when (a) and (b) have the same parity. Hence the equivalence classes are the odd class ({1,3}) and the even class ({2,4}).
