GATE CS 2021 Set 2 — Question 21
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Engineering Mathematics → Sets & Combinatorics → Sets, Relations & Functions
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Question
Consider the following sets, where :
: Set of all matrices with entries from the set
: Set of all functions from the set to the set
Which of the following choice(s) is/are correct?
: Set of all matrices with entries from the set
: Set of all functions from the set to the set
Which of the following choice(s) is/are correct?
Correct answer
(B) There exists a surjection from S₁ to S₂.; (C) There exists a bijection from S₁ to S₂.
Solution
Let us calculate the cardinality of sets and .For :
is the set of all matrices with entries from .
An matrix has entries.
Each entry can be chosen in 3 ways (either , , or ).
Therefore, the total number of such matrices is .For :
is the set of all functions from domain to codomain .
The size of the domain is .
The size of the codomain is .
The total number of functions is .
Therefore, .Conclusion:
Since , the two finite sets have the same cardinality. Thus, there exists a bijection between them.
Since a bijection exists, it implies that there exists an injection and there exists a surjection from to .Evaluating the options:
(A) False. A bijection exists.
(B) True. A surjection exists.
(C) True. A bijection exists.
(D) False. An injection exists.Correct options are (B) and (C).
is the set of all matrices with entries from .
An matrix has entries.
Each entry can be chosen in 3 ways (either , , or ).
Therefore, the total number of such matrices is .For :
is the set of all functions from domain to codomain .
The size of the domain is .
The size of the codomain is .
The total number of functions is .
Therefore, .Conclusion:
Since , the two finite sets have the same cardinality. Thus, there exists a bijection between them.
Since a bijection exists, it implies that there exists an injection and there exists a surjection from to .Evaluating the options:
(A) False. A bijection exists.
(B) True. A surjection exists.
(C) True. A bijection exists.
(D) False. An injection exists.Correct options are (B) and (C).
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