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
There are infinitely many 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
We are given a set S consisting of disjoint intervals and a set T . Let’s evaluate each statement:
- Statement (A): There are infinitely many functions from S to T . This is true because we can define functions that map each element in S to any element in T , which results in infinitely many possible functions.
- Statement (B): There are infinitely many strictly increasing functions from S to T . This is not necessarily true. Since T is a finite set, only a limited number of strictly increasing functions can be formed based on the available values in T.
- Statement (C): The number of continuous functions from S to T is at most 120. This is true because T has only 4 elements and continuity limits the number of distinct ways we can map intervals to these elements. Hence, there are at most 120 continuous functions.
- Statement (D): Every continuous function from S to T is differentiable. This is true because any continuous function defined on intervals from a finite set T will necessarily be differentiable.
Thus, the correct answer is (A), (C), (D).
Quick Tip
When dealing with sets and mappings, consider the constraints on continuity and differentiability when choosing the correct answers.