Let ƒ : A → B be a function. Prove that f is onto if and only if there exists a function g : B → A so that f og= 1B.
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A: We will solve the problem using the definition of onto function.
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Q: Determine whether or not the function f : Z × (Z − {0}) → Z is onto, if f((m, n)) = ⌊m/n ⌋
A: Given that: f(m,n)=mn, f:Zx(Z-{0})→Z
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- 27. Let , where and are nonempty. Prove that has the property that for every subset of if and only if is one-to-one. (Compare with Exercise 15 b.). 15. b. For the mapping , show that if , then .Let g:AB and f:BC. Prove that f is onto if fg is onto.10. Let and be mappings from to. Prove that if is invertible, then is onto and is one-to-one.
- Label each of the following statements as either true or false. A mapping is onto if and only if its codomain and range are equal.28. Let where and are nonempty. Prove that has the property that for every subset of if and only if is onto. (Compare with Exercise 15c.) Exercise 15c. c. For this same and show that.[Type here] 18. Prove that only idempotent elements in an integral domain are and . [Type here]
- Let f:AB and g:BA. Prove that f is one-to-one and onto if fg is one to-one and gf onto.Let a and b be constant integers with a0, and let the mapping f:ZZ be defined by f(x)=ax+b. Prove that f is one-to-one. Prove that f is onto if and only if a=1 or a=1.For each of the following parts, give an example of a mapping from E to E that satisfies the given conditions. a. one-to-one and onto b. one-to-one and not onto c. onto and not one-to-one d. not one-to-one and not onto