If the set x contains 7 elements and set y contains 8 elements, then the number of bijections from x to y is
- 0
- 7!
- \(^{8}P_{7}\)
- 8!
The Correct Option is A
Approach Solution - 1
Given:
Set X has 7 elements and Set Y has 8 elements.
To find: The number of bijections from X to Y.
Concept:
A bijection is a function that is both one-to-one (injective) and onto (surjective).
For a bijection to exist between two sets, they must have the same number of elements.
Since |X| = 7 and |Y| = 8, and 7 ≠ 8, it is not possible to form a bijection between X and Y.
Answer: 0
Approach Solution -2
A bijection is a function that is both injective (one-to-one) and surjective (onto).
- Injective (one-to-one): Each element in the domain maps to a unique element in the codomain.
- Surjective (onto): Every element in the codomain has a pre-image in the domain (i.e., every element in the codomain is mapped to by at least one element in the domain).
For a bijection to exist between two sets, they must have the same number of elements. In other words, |X| = |Y|.
In this problem, set X has 7 elements, and set Y has 8 elements. Since the number of elements is different (|X| ≠ |Y|), it is impossible to create a bijection between the two sets.
Therefore, the number of bijections from X to Y is 0.
Answer: 0
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