Question

Which of the following statements are true? Select all that applies (check all that apply)A.

Let f:Z→{0,1,2,3,4,5} be the map sending x to its remainder when divided by 6 .
B..Let S={0,1,2,3},T={2,3,4,5,6}. Then f(S∩T)=f(S)∩f(T)
C..Let f:{1,2,3,4,5}→{1,2,3,4,5} be the map defined by f(i)=i+1,1≤i≤4,f(5)=1.
D.Let S={1,2,3},T={3,4,5}. Then f(S∩T)=f(S)∩f(T).

Answer 1

The evaluated statements A, B, C, and D involving functions and set theory are all true. Statements A and C are definitions and naturally true, while statements B and D follow the principles of functions and set operations correctly.

Step-by-step explanation:

The statements in question are referring to functions and set operations. For statement A, the function f maps any integer x to its remainder when divided by 6, which indeed produces a result that belongs to the set {0,1,2,3,4,5}. Therefore, statement A is true.

For statement B, we must evaluate f(S∩T) and f(S)∩f(T) where S={0,1,2,3} and T={2,3,4,5,6}. The intersection of S and T is {2,3}, so f(S∩T) = {2,3}. However, f(S) = {0,1,2,3} and f(T) = {2,3,4,5,0} (since 6 modulo 6 is 0), so f(S)∩f(T) = {2,3}. Thus, f(S∩T) equals f(S)∩f(T), making statement B true.

Statement C defines a function f from {1,2,3,4,5} to itself with f(i)=i+1 for 1≤i≤4 and f(5)=1. This statement is simply a definition true by its own terms.

For statement D, with S={1,2,3} and T={3,4,5}, and using the function defined in statement C, we have f(S∩T) = f({3}) = {4}, while f(S) = {2,3,4} and f(T) = {4,5,1}. Hence, f(S)∩f(T) = {4}. Thus, f(S∩T) equals f(S)∩f(T), and statement D is also true.