Let S = {1, 2, 3, 4}.Then the number of elements in the set {f : S × S → S : f is onto and f(a,b)=f(b,a) ≥ a ∀ (a, b)∈ S × S is _____.
Correct Answer: 37
Solution and Explanation
The correct answer is 37
There are 16 ordered pairs in S × S. We write all these ordered pairs in 4 sets as follows.
A = {(1, 1)}
B = {(1, 4), (2, 4), (3, 4) (4, 4), (4, 3), (4, 2), (4, 1)}
C = {(1, 3), (2, 3), (3, 3), (3, 2), (3, 1)}
D = {(1, 2), (2, 2), (2, 1)}
All elements of set B have image 4 and only element of A has image 1.
All elements of set C have image 3 or 4 and all elements of set D have image 2 or 3 or 4.
We will solve this question in two cases.
Case I : When no element of set C has image 3.
Number of onto functions = 2 (when elements of set D have images 2 or 3)
Case II : When atleast one element of set C has image 3.
Number of onto functions = (23 – 1)(1 + 2 + 2) = 35
Therefore , total number of functions = 37
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Concepts Used:
Sets
In mathematics, a set is a well-defined collection of objects. Sets are named and demonstrated using capital letter. In the set theory, the elements that a set comprises can be any sort of thing: people, numbers, letters of the alphabet, shapes, variables, etc.
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Elements of a Set:
The items existing in a set are commonly known to be either elements or members of a set. The elements of a set are bounded in curly brackets separated by commas.
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Cardinal Number of a Set:
The cardinal number, cardinality, or order of a set indicates the total number of elements in the set.
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