Let S = (0,1) ∪ (1,2) ∪ (3,4) and T = {0,1, 2,3} . Then which of the following statements is(are) true?
- There are infinitely many functions from S to T
- There are infinitely many strictly increasing functions from S to T
- The number of continuous functions from S to T is at most 120
- Every continuous function from S to T is differentiable
The Correct Option is A, C, D
Approach Solution - 1
(1) There are infinitely many functions from S to T: This statement is true. Since the cardinality of set T is greater than the cardinality of set S, there are infinitely many possible mappings from S to T. Each element in S can be mapped to any element in T, allowing for a variety of different functions.
(2) There are infinitely many strictly increasing functions from S to T. This statement is false. Since the cardinality of T is finite, the number of strictly increasing functions from S to T is also finite. The size of the set T limits the number of possible strictly increasing mappings.
(3) The number of continuous functions from S to T is at most 120. This statement is true. To determine the number of continuous functions from S to T, we need to consider the number of possible mappings from the intervals in S to the elements in T while maintaining continuity. Each interval in S can be mapped to any subset of T containing one or more elements. Since there are three intervals in S, we have a total of 3 choices for mapping each interval. Therefore, the total number of continuous functions from S to T is at most 3^3 = 27, which is less than 120.
(4) There are many ways to assign a value of T to elements of domain, hence infinitely many functions will exist from set S to set T.
Approach Solution -2
Given :
S = (0, 1) ∪ (1, 2) ∪ (3, 4) and T = {0, 1, 2, 3}.
Let S be the domain and T be the co-domain of the function y = f(x).
Let's verify all the options :
(A) The domain S contains infinitely many elements, while the co-domain T has four elements.
There are infinitely many possible functions from S to T.
So, the option (A) is correct.
(B) If the number of elements in the domain is greater than the number of elements in the co-domain, then the number of strictly increasing functions is zero.
So, the option (B) is incorrect.
(C) The maximum number of continuous functions is \(4 \times 4 \times 4 = 64\), as each interval (0, 1), (1, 2), and (3, 4) has four possible choices.
∵ 64 < 120
So, the option (C) is correct.
(D) For every point where f(x) is continuous, f'(x) = 0. Every continuous function from S to T is differentiable.
So, the option (D) is correct.
Therefore, the correct options are (A), (C) and (D).
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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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