Relations

An ordered pair consists of two objects or elements in a fixed order. It is written in the form (a, b) where, a is called the ‘first component’ and b is called the ‘second component’.
The cartesian product of two non-empty sets A and B is defined as the set of all ordered pairs (a, b) where a ? A and b ? B. It is denoted by A ´ B.
Let A and B be two sets. Then a relation R from set A to set B is a subset of A ´ B, if R is a relation from A to B then R ? A ´ B.
The set of all first components of the ordered pairs, in a relation R, from a set A to a set B is called the domain of relation R.
The set of all second components of the ordered pairs, in a relation R, from a set A to a set B is called the range of relation R.
For any relation R from a set A to set B, the whole set B is called the codomain of the relation R.
A relation R in a set A is called universal relation, if each element of A is related to every element of A, i.e., R = A x A.
A relation R in a set A is called empty relation, if no element of A is related to any element of A, i.e., R = ?? ? A x A.
Both the empty relation and the universal relation are called trivial relations.
A relation R in a set A is called identity relation if each element in A is related to itself only. i.e., R = {(a, a): ? a ? A.
Let R be a relation on a non-empty set A then R is called a reflexive relation iff a R a, i.e., (a, a) ? R for all a ? A.
Let A be a set and R be a relation defined on A. If (x, y) ? R ? (y, x) ? R, then R is called a symmetric relation.
Let R be a relation on a non-empty set A. Then R is called a transitive relation if (a, b) ? R, (b, c) ? R ? (a, c) ? R for all a, b, c ? A.
A relation R in a set A is said to be an equivalence relation iff R is reflective, symmetric and transitive.
Keywords: Congruence Modulo m, Equivalence classes

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  • Q1

    If A x B = {(a, 1), (b, 3), (a, 3), (b, 1), (a, 2), (b, 2)}, then the value of A and B are respectively equal to

    Marks:1
    Answer:

    {a, b}, {1, 2, 3}.

    Explanation:

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  • Q2

    Marks:1
    Answer:

    {(1, 3), (2, 3), (3, 3)}.

    Explanation:
     
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  • Q3

    Marks:1
    Answer:

    Explanation:

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  • Q4

    If A is the set of even natural numbers less than 8 and B is the set of prime numbers less than 7, then the number of relations from A to B is

    Marks:1
    Answer:

    29

    Explanation:

    A = {2, 4, 6 }, B = { 2, 3, 5}

    No, of elements in the cartesian product A χ B = 3 × 3 = 9

    So, the number of relations from set A to B is 29

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  • Q5

    If the set A has 3 elements and the set b = {3,4,5} then find the number of elements in (A × B).

    Marks:1
    Answer:

    9

    Explanation:
    Set A has 3 elements and set B also has 3 elements. Then, number of elements in A × B = 3 × 3 = 9
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