GATE CS 2024 Set 1 — Question 32
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Engineering Mathematics → Sets & Combinatorics → Sets, Relations & Functions
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Question
Let and be non-empty finite sets such that there exist one-to-one and onto functions (i) from to and (ii) from to . The number of possible values of is __________
Correct answer
2 to 2
Solution
Let and .From condition (i), there is a bijection from to , which implies . Thus, .From condition (ii), there is a bijection from to , which implies .We know that:
Equating the two:
Rearranging for :
We know that for any two sets and , the size of their intersection satisfies:
Substituting into the inequality:
Since is a positive integer (non-empty set), this implies .
This implies .Combining these, the possible integer values for are and .Verification:
Equating the two:
Rearranging for :
We know that for any two sets and , the size of their intersection satisfies:
Substituting into the inequality:
1.
Since is a positive integer (non-empty set), this implies .
2.
This implies .Combining these, the possible integer values for are and .Verification:
- If : . This is possible if (e.g., ).
- If : . This is possible if and are disjoint (e.g., ).
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