All but two of the following statements are correct ways to express the fact that a function f is onto. Find the two that are incorrect.
a. f is onto
b. f is onto
c. f is onto
d. f is onto
e. f is onto
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Chapter 7 Solutions
Discrete Mathematics With Applications
- 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 ontoarrow_forwardLet g:AB and f:BC. Prove that f is onto if fg is onto.arrow_forwardLet 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.arrow_forward
- 10. Let and be mappings from to. Prove that if is invertible, then is onto and is one-to-one.arrow_forwardFor each of the following mappings f:ZZ, determine whether the mapping is onto and whether it is one-to-one. Justify all negative answers. a. f(x)=2x b. f(x)=3x c. f(x)=x+3 d. f(x)=x3 e. f(x)=|x| f. f(x)=x|x| g. f(x)={xifxiseven2x1ifxisodd h. f(x)={xifxisevenx1ifxisodd i. f(x)={xifxisevenx12ifxisodd j. f(x)={x1ifxiseven2xifxisoddarrow_forwardGive an example of mappings and such that one of or is not onto but is onto.arrow_forward
- Label each of the following statements as either true or false. Let R be a relation on a nonempty set A that is symmetric and transitive. Since R is symmetric xRy implies yRx. Since R is transitive xRy and yRx implies xRx. Hence R is alsoreflexive and thus an equivalence relation on A.arrow_forwardLabel each of the following statements as either true or false. A mapping is onto if and only if its codomain and range are equal.arrow_forwardTrue or False Label each of the following statements as either true or false. Let and such that is onto. Then both and are onto.arrow_forward
- 4. Let , where is nonempty. Prove that a has left inverse if and only if for every subset of .arrow_forward5. For each of the following mappings, determine whether the mapping is onto and whether it is one-to-one. Justify all negative answers. (Compare these results with the corresponding parts of Exercise 4.) a. b. c. d. e. f.arrow_forward27. 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 .arrow_forward
- Elements Of Modern AlgebraAlgebraISBN:9781285463230Author:Gilbert, Linda, JimmiePublisher:Cengage Learning,