Let the number of elements in sets \(A\) and \(B\) be five and two respectively. Then the number of subsets of \(A \times B\) each having at least 3 and at most 6 elements is:
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752
772
792
782
The Correct Option is C
Solution and Explanation
Step 1: Find the total number of elements in \(A \times B\).
- The Cartesian product \(A \times B\) contains \(n(A) \times n(B) = 5 \times 2 = 10\) elements.
Step 2: Calculate the subsets.
- The total number of subsets of \(A \times B\) is \(2^{10} = 1024\).
- Subsets with at least 3 and at most 6 elements are given by: \[ \binom{10}{3} + \binom{10}{4} + \binom{10}{5} + \binom{10}{6}. \] - Calculate each term: \[ \binom{10}{3} = 120, \quad \binom{10}{4} = 210, \quad \binom{10}{5} = 252, \quad \binom{10}{6} = 210. \] - Summing these: \[ 120 + 210 + 252 + 210 = 792. \] Final Answer: The number of subsets is \(792\).
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Concepts Used:
Types of Relation
TYPES OF RELATION
Empty Relation
Relation is said to be empty relation if no element of set X is related or mapped to any element of X i.e, R = Φ.
Universal Relation
A relation R in a set, say A is a universal relation if each element of A is related to every element of A.
R = A × A.
Identity Relation
Every element of set A is related to itself only then the relation is identity relation.
Inverse Relation
Let R be a relation from set A to set B i.e., R ∈ A × B. The relation R-1 is said to be an Inverse relation if R-1 from set B to A is denoted by R-1
Reflexive Relation
If every element of set A maps to itself, the relation is Reflexive Relation. For every a ∈ A, (a, a) ∈ R.
Symmetric Relation
A relation R is said to be symmetric if (a, b) ∈ R then (b, a) ∈ R, for all a & b ∈ A.
Transitive Relation
A relation is said to be transitive if, (a, b) ∈ R, (b, c) ∈ R, then (a, c) ∈ R, for all a, b, c ∈ A
Equivalence Relation
A relation is said to be equivalence if and only if it is Reflexive, Symmetric, and Transitive.