Q.Traffic flows from π· to πΈ and π· to πΆ. The department wants to represent and analyze this data using relations and functions. Use the given data to answer the following questions: I. Is the traffic flow reflexive? Justify. [1] II. Is the traffic flow transitive? Justify. [1] III
(A) Represent the relation describing the traffic flow as a set of ordered pairs. Also state the domain and range of the relation.
(B) Does the traffic flow represent a function? Justify your answer. [2] 4
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Start your 14-day free trial to unlock the full solution βPart (a)Concept understanding β Relation Properties
Properties of a Relation
A relation on a set pairs elements of with one another. Some relations behave in regular, predictable ways, and we name these behaviours properties. Three matter most for CBSE Class 12 β reflexive, symmetric, transitive (together they build an equivalence relation); a fourth, antisymmetric, is worth knowing for order relations.
Reflexive β everything relates to itself
is reflexive if for every . "Has the same age as" is reflexive; "is taller than" is not. If even one element misses its self-pair, reflexivity fails: on , is not reflexive because is absent.
Symmetric β the relation runs both ways
is symmetric if . "Is married to" is symmetric; "is taller than" is not. Symmetry does not demand that every pair be related β only that any pair which appears also appears reversed. So is symmetric, but is not, since is missing.
Transitive β relations chain
is transitive if and together force . "Is an ancestor of" is transitive; "is a friend of" is not. A single broken chain breaks the property: is not transitive because is missing.
Antisymmetric β two-way ties force equality
is antisymmetric if and together force . The order relation is antisymmetric: and give . It does not ban self-pairs like ; it only rules out distinct elements related both ways.
Test the properties in order of ease β reflexivity first, then symmetry, transitivity. A single counterexample is enough to disprove any of them.
| Property | Condition |
|---|---|
| Reflexive |
Part (b)Concept understanding β Relation Properties
Properties of a Relation
A relation on a set pairs elements of with one another. Some relations behave in regular, predictable ways, and we name these behaviours properties. Three matter most for CBSE Class 12 β reflexive, symmetric, transitive (together they build an equivalence relation); a fourth, antisymmetric, is worth knowing for order relations.
Reflexive β everything relates to itself
is reflexive if for every . "Has the same age as" is reflexive; "is taller than" is not. If even one element misses its self-pair, reflexivity fails: on , is not reflexive because is absent.
Symmetric β the relation runs both ways
is symmetric if . "Is married to" is symmetric; "is taller than" is not. Symmetry does not demand that every pair be related β only that any pair which appears also appears reversed. So is symmetric, but is not, since is missing.
Transitive β relations chain
is transitive if and together force . "Is an ancestor of" is transitive; "is a friend of" is not. A single broken chain breaks the property: is not transitive because is missing.
Antisymmetric β two-way ties force equality
is antisymmetric if and together force . The order relation is antisymmetric: and give . It does not ban self-pairs like ; it only rules out distinct elements related both ways.
Test the properties in order of ease β reflexivity first, then symmetry, transitivity. A single counterexample is enough to disprove any of them.
| Property | Condition |
|---|---|
| Reflexive |
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