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Let f be a function from a set A to a set B, g a function from B to C, and h a function from A to C, such that h(a) = g(f(a)) for all a ∈ A. Which of the following statements is always true for all such functions f and g?
- a)g is onto => h is onto
- b)h is onto => f is onto
- c)h is onto => g is onto
- d)h is onto => f and g are ont
Correct answer is option 'C'. Can you explain this answer?
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Most Upvoted Answer
Let f be a function from a set A to a set B, g a function from B to C,...
Explanation:
To prove that the statement "h is onto = g is onto" is always true for all functions f and g, we need to show that if h is onto, then g is onto, and vice versa.
Proof:
1. Suppose h is onto. This means that for every element c in set C, there exists an element a in set A such that h(a) = c.
2. Let's consider an arbitrary element c in set C. Since h is onto, there exists an element a in set A such that h(a) = c.
3. According to the definition of h, we have h(a) = g(f(a)). So, c = g(f(a)).
4. This implies that for every element c in set C, there exists an element b = f(a) in set B such that g(b) = c.
5. Therefore, g is onto.
Proof (continued):
1. Suppose g is onto. This means that for every element c in set C, there exists an element b in set B such that g(b) = c.
2. Let's consider an arbitrary element c in set C. Since g is onto, there exists an element b in set B such that g(b) = c.
3. According to the definition of g, there exists an element a = f^(-1)(b) in set A such that f(a) = b.
4. This implies that for every element c in set C, there exists an element a = f^(-1)(b) in set A such that h(a) = g(f(a)) = g(b) = c.
5. Therefore, h is onto.
Conclusion:
From the two proofs above, we can conclude that if h is onto, then g is onto, and vice versa. Therefore, the statement "h is onto = g is onto" is always true for all functions f and g.
To prove that the statement "h is onto = g is onto" is always true for all functions f and g, we need to show that if h is onto, then g is onto, and vice versa.
Proof:
1. Suppose h is onto. This means that for every element c in set C, there exists an element a in set A such that h(a) = c.
2. Let's consider an arbitrary element c in set C. Since h is onto, there exists an element a in set A such that h(a) = c.
3. According to the definition of h, we have h(a) = g(f(a)). So, c = g(f(a)).
4. This implies that for every element c in set C, there exists an element b = f(a) in set B such that g(b) = c.
5. Therefore, g is onto.
Proof (continued):
1. Suppose g is onto. This means that for every element c in set C, there exists an element b in set B such that g(b) = c.
2. Let's consider an arbitrary element c in set C. Since g is onto, there exists an element b in set B such that g(b) = c.
3. According to the definition of g, there exists an element a = f^(-1)(b) in set A such that f(a) = b.
4. This implies that for every element c in set C, there exists an element a = f^(-1)(b) in set A such that h(a) = g(f(a)) = g(b) = c.
5. Therefore, h is onto.
Conclusion:
From the two proofs above, we can conclude that if h is onto, then g is onto, and vice versa. Therefore, the statement "h is onto = g is onto" is always true for all functions f and g.
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Let f be a function from a set A to a set B, g a function from B to C, and h a function from A to C, such that h(a) = g(f(a)) for all a A. Which of the following statements is always true for all such functions f and g? a)g is onto = h is ontob)h is onto = f is ontoc)h is onto = g is ontod)h is onto = f and g are ontCorrect answer is option 'C'. Can you explain this answer?
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Let f be a function from a set A to a set B, g a function from B to C, and h a function from A to C, such that h(a) = g(f(a)) for all a A. Which of the following statements is always true for all such functions f and g? a)g is onto = h is ontob)h is onto = f is ontoc)h is onto = g is ontod)h is onto = f and g are ontCorrect answer is option 'C'. 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 be a function from a set A to a set B, g a function from B to C, and h a function from A to C, such that h(a) = g(f(a)) for all a A. Which of the following statements is always true for all such functions f and g? a)g is onto = h is ontob)h is onto = f is ontoc)h is onto = g is ontod)h is onto = f and g are ontCorrect answer is option 'C'. 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 be a function from a set A to a set B, g a function from B to C, and h a function from A to C, such that h(a) = g(f(a)) for all a A. Which of the following statements is always true for all such functions f and g? a)g is onto = h is ontob)h is onto = f is ontoc)h is onto = g is ontod)h is onto = f and g are ontCorrect answer is option 'C'. Can you explain this answer?.
Let f be a function from a set A to a set B, g a function from B to C, and h a function from A to C, such that h(a) = g(f(a)) for all a A. Which of the following statements is always true for all such functions f and g? a)g is onto = h is ontob)h is onto = f is ontoc)h is onto = g is ontod)h is onto = f and g are ontCorrect answer is option 'C'. 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 be a function from a set A to a set B, g a function from B to C, and h a function from A to C, such that h(a) = g(f(a)) for all a A. Which of the following statements is always true for all such functions f and g? a)g is onto = h is ontob)h is onto = f is ontoc)h is onto = g is ontod)h is onto = f and g are ontCorrect answer is option 'C'. 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 be a function from a set A to a set B, g a function from B to C, and h a function from A to C, such that h(a) = g(f(a)) for all a A. Which of the following statements is always true for all such functions f and g? a)g is onto = h is ontob)h is onto = f is ontoc)h is onto = g is ontod)h is onto = f and g are ontCorrect answer is option 'C'. Can you explain this answer?.
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