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Let f: B → C and g: A → B be two functions and let h = f o g. Given that h is an onto function. Which one of the following is TRUE?
- a)f and g should both be onto functions.
- b)f should be onto but g need not be onto
- c)g should be onto but f need not be onto
- d)both f and g need not be onto
Correct answer is option 'B'. Can you explain this answer?
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Verified Answer
Let f: B → C and g: A → B be two functions and let h = f o...
A function f: X → Y is called on-to function if for every value in set Y, there is a value in set X.
Given that, f: B → C and g: A → B and h = f o g.
Note that the sign o represents composition.
h is basically f(g(x)). So h is a function from set A to set C.
It is also given that h is an onto function which means for every value in C there is a value in A.
Note that the sign o represents composition.
h is basically f(g(x)). So h is a function from set A to set C.
It is also given that h is an onto function which means for every value in C there is a value in A.
We map from C to A using B. So for every value in C, there must be a value in B. It means f must be onto. But g may or may not be onto as there may be some values in B which don't map to A.
Example :
Let us consider following sets
A : {a1, a2, a3}
B : {b1, b2}
C : {c1} And following function values
f(b1) = c1
g(a1) = b1, g(a2) = b1, g(a3) = b1
Values of h() would be, h(a1) = c1, h(a2) = c1, h(a3) = c1
Here h is onto, therefore f is onto, but g is onto as b2 is not mapped to any value in A.
A : {a1, a2, a3}
B : {b1, b2}
C : {c1} And following function values
f(b1) = c1
g(a1) = b1, g(a2) = b1, g(a3) = b1
Values of h() would be, h(a1) = c1, h(a2) = c1, h(a3) = c1
Here h is onto, therefore f is onto, but g is onto as b2 is not mapped to any value in A.
Given that, f: B → C and g: A → B and h = f o g.
Most Upvoted Answer
Let f: B → C and g: A → B be two functions and let h = f o...
Explanation:
Given:
- We have two functions, f: B -> C and g: A -> B.
- We define the function h = f o g, which means h(x) = f(g(x)) for all x in A.
To prove:
- If h is an onto function, then f should be onto but g need not be onto.
Proof:
To prove this statement, we need to consider the definition of an onto function and analyze the functions f, g, and h.
Onto Function:
- An onto function, also known as a surjective function, is a function where every element in the co-domain has at least one pre-image in the domain.
- In other words, for every y in the co-domain, there exists an x in the domain such that f(x) = y.
Analysis:
1. We have h(x) = f(g(x)) for all x in A.
2. Since h is an onto function, for every y in the co-domain of h, there exists an x in A such that h(x) = y.
3. Let's consider an arbitrary y in the co-domain of h.
4. Then there exists an x in A such that h(x) = y.
5. By the definition of h, we have h(x) = f(g(x)).
6. So, y = h(x) = f(g(x)).
7. This implies that for every y in the co-domain of h, there exists an x in A such that f(g(x)) = y.
8. Therefore, for every y in the co-domain of h, there exists an x in A such that f(x) = y, where x = g(x).
9. Hence, we can conclude that f is an onto function.
Conclusion:
- If h is an onto function, then f should be onto but g need not be onto.
- Therefore, option 'B' is the correct answer.
Given:
- We have two functions, f: B -> C and g: A -> B.
- We define the function h = f o g, which means h(x) = f(g(x)) for all x in A.
To prove:
- If h is an onto function, then f should be onto but g need not be onto.
Proof:
To prove this statement, we need to consider the definition of an onto function and analyze the functions f, g, and h.
Onto Function:
- An onto function, also known as a surjective function, is a function where every element in the co-domain has at least one pre-image in the domain.
- In other words, for every y in the co-domain, there exists an x in the domain such that f(x) = y.
Analysis:
1. We have h(x) = f(g(x)) for all x in A.
2. Since h is an onto function, for every y in the co-domain of h, there exists an x in A such that h(x) = y.
3. Let's consider an arbitrary y in the co-domain of h.
4. Then there exists an x in A such that h(x) = y.
5. By the definition of h, we have h(x) = f(g(x)).
6. So, y = h(x) = f(g(x)).
7. This implies that for every y in the co-domain of h, there exists an x in A such that f(g(x)) = y.
8. Therefore, for every y in the co-domain of h, there exists an x in A such that f(x) = y, where x = g(x).
9. Hence, we can conclude that f is an onto function.
Conclusion:
- If h is an onto function, then f should be onto but g need not be onto.
- Therefore, option 'B' is the correct answer.
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Let f: B → C and g: A → B be two functions and let h = f o g. Given that h is an onto function. Which one of the following is TRUE?a)f and g should both be onto functions.b)f should be onto but g need not be ontoc)g should be onto but f need not be ontod)both f and g need not be ontoCorrect answer is option 'B'. Can you explain this answer?
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Let f: B → C and g: A → B be two functions and let h = f o g. Given that h is an onto function. Which one of the following is TRUE?a)f and g should both be onto functions.b)f should be onto but g need not be ontoc)g should be onto but f need not be ontod)both f and g need not be ontoCorrect answer is option 'B'. Can you explain this answer? for Computer Science Engineering (CSE) 2024 is part of Computer Science Engineering (CSE) preparation. The Question and answers have been prepared according to the Computer Science Engineering (CSE) exam syllabus. Information about Let f: B → C and g: A → B be two functions and let h = f o g. Given that h is an onto function. Which one of the following is TRUE?a)f and g should both be onto functions.b)f should be onto but g need not be ontoc)g should be onto but f need not be ontod)both f and g need not be ontoCorrect answer is option 'B'. Can you explain this answer? covers all topics & solutions for Computer Science Engineering (CSE) 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Let f: B → C and g: A → B be two functions and let h = f o g. Given that h is an onto function. Which one of the following is TRUE?a)f and g should both be onto functions.b)f should be onto but g need not be ontoc)g should be onto but f need not be ontod)both f and g need not be ontoCorrect answer is option 'B'. Can you explain this answer?.
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