Answer :
f is a function from set A to set B, and let's check the statements:
- Statement a is indeterminate without more information about the function f.
- Statement b is indeterminate without more information about the function f.
- Statement c is true only if set A is equal to set B.
- Statement d is true if the function f is both one-to-one and onto.
- Statement e is chosen if none of the previous statements are true.
a. If f is one-to-one: This means that each element in set A maps to a unique element in set B. To check if this statement is true, we need more information about the function f. Without additional details, we cannot determine if this statement is true or false.
b. If f is onto: This means that every element in set B has a corresponding element in set A. In other words, the range of the function f is equal to set B. If this statement is true, it means that the function f is surjective. Again, without more information about f, we cannot determine if this statement is true or false.
c. If A = B: This statement is saying that set A is equal to set B. If this is the case, it means that every element in set A maps to an element in set A itself. In other words, every element in set A is its own image. This statement is only true if set A and set B are the same.
d. If f is one-to-one and onto: This statement combines both conditions. If f is both one-to-one and onto, it means that each element in set A maps to a unique element in set B, and every element in set B has a corresponding element in set A. This statement is true only if both conditions hold.
e. None of the above: This option is chosen if none of the previous statements (a, b, c, or d) are true.
In summary:
- Statement a is indeterminate without more information about the function f.
- Statement b is indeterminate without more information about the function f.
- Statement c is true only if set A is equal to set B.
- Statement d is true if the function f is both one-to-one and onto.
- Statement e is chosen if none of the previous statements are true.
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