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: A, B and C
Explanation:
(A) There are infinitely many functions from S to T:
To understand this, let's think about each real number in the intervals (0, 1), (1, 2), and (3, 4) as a point in S. For each of these points, we have 4 choices to map it to in T: 0, 1, 2, or 3. Since there are infinitely many real numbers in each interval, there are infinitely many possible mappings from S to T.
(B) There are infinitely many strictly increasing functions from S to T:
A function is strictly increasing if it preserves the order of numbers. In our case, we have intervals in S, and we can map points from these intervals to points in T in an increasing order. Since there are infinitely many ways to arrange points in S to be strictly increasing, 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:
Continuous functions from S to T are determined by how points in the intervals (0, 1), (1, 2), and (3, 4) are mapped to points in T. For each interval, we have 4 choices of points to map to. Since there are 3 disjoint intervals, the total number of continuous functions is the product of these choices, which is 4 x 4 x 4 = 64. Thus, there are at most 64 continuous functions from S to T, not 120.
Therefore, the corrected set of true statements is (A), (B), and (C).
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