Answer :
Final answer:
The correct statements are A (fog onto implies f onto), C (fog onto and g not onto implies f not 1-1), and E (f and g onto implies fog onto).
Explanation:
In this question, we are given two functions:
- f:B + C
- g: A + B
We need to check which statements are true:
- A. If fog is onto, then so is f.
- To determine if this statement is true, we need to analyze the composition of functions. If fog is onto, it means for every element in C, there exists an element in A that maps to it. Since f:B + C, it means that for every element in B, there exists an element in C that maps to it. Therefore, f is also onto.
- So statement A is true.
- B. If fog is onto, then so is g.
- This statement is not necessarily true. The onto mapping in fog does not guarantee that every element in A is mapped to by g.
- So statement B is false.
- C. If fog is onto and g is not, then f cannot be 1-1.
- This statement is true. If g is not onto, it means there exists an element in B that is not mapped to by g. When we compose with f, there will be elements in B that are not mapped to by fog, making fog not onto. Therefore, f cannot be 1-1.
- So statement C is true.
- D. If fog is onto and f is not, then g cannot be 1-1.
- This statement is not necessarily true. The composition of functions fog being onto does not provide enough information about the 1-1 property of g.
- So statement D is false.
- E. If f and g are both onto, then so is fog.
- This statement is true. If f and g are both onto, it means that for every element in B, there exists an element in C that maps to it (by f) and for every element in A, there exists an element in B that maps to it (by g). Therefore, by composing f and g, fog will be onto.
- So statement E is true.
- F. None of the above.
- This statement is false as some of the statements above are true. So
- statement F is false.
Therefore, the correct statements are A (fog onto implies f onto), C (fog onto and g not onto implies f not 1-1), and E (f and g onto implies fog onto).
Learn more about Composition of Functions here:
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Final answer:
The correct statements are B and D: if fog is onto, then so is g, and if fog is onto and f is not, then g cannot be 1-1.
Explanation:
To determine which statements are true, let's analyze each one:
A. If fog is onto then so is f.
This statement is not necessarily true. If g is not onto, then fog cannot be onto regardless of whether f is onto or not.
B. If fog is onto, then so is g.
This statement is true. If fog is onto, it means that for every element c in C, there exists an element a in A such that fog(a) = c. Since g:B + C, g must also be onto.
C. If fog is onto and g is not, then f cannot be 1-1.
This statement is not necessarily true. The onto-ness or not-onto-ness of g does not necessarily affect the one-to-one property of f. There can be cases where fog is onto, g is not onto, and f is still one-to-one.
D. If fog is onto and f is not, then g cannot be 1-1.
This statement is true. If fog is onto, it means for every element c in C, there exists an element a in A such that fog(a) = c. If f is not one-to-one, it means there exists two distinct elements a1 and a2 in A such that f(a1) = f(a2). Since fog(a1) = fog(a2) = c for some c in C, it implies that a1 and a2 both map to the same element c in C. Thus, g cannot be one-to-one because the same element c in C is being mapped to by two different elements a1 and a2 in A.
E. If f and g are both onto, then so is fog.
This statement is true. If f and g are both onto, it means that for every element c in C, there exists an element b in B such that f(b) = c. And for every element b in B, there exists an element a in A such that g(a) = b. Combining these two properties, for every element c in C, there exists an element a in A such that fog(a) = c. Hence, fog is onto.
F. None of the above.
This statement is not true since statements B and D are true. Therefore, the correct statements are B and D.
