Question

Let S = (0,1) ∪ (1,2) ∪ (3,4) and T = {0,1, 2,3} . Then which of the following statements is(are) true?

• A. There are infinitely many functions from S to T
• B. There are infinitely many strictly increasing functions from S to T
• C. The number of continuous functions from S to T is at most 120
• D. Every continuous function from S to T is differentiable

Answer 1

Answer: (A), (C), (D)

There are infinitely many functions from S to T

Answer 2

(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.