Module 07 Short Answer Assignment - Functions, Domains, Ranges, 1-1, Onto - Answer Key Each graph represents a relation (A) F : X Y , that may also be a function. (B) X Y 1 (C) X (D) Y X Y 1X aY 1 a 1 a 1 a 2 b 2 b 2 b 2 b 3 c 3 c 3 c 3 c 4 d 4 d 4 d 4 d (E) (F) X Y X (G) X (H) Y X Y Y 1 a 1 a 1 a 1 a 2 b 2 b 2 b 2 b 3 c 3 c 3 c 3 c 4 d 4 d 4 d 4 d For questions 1 - 6, answer the following questions by typing the number above the graph on the line. If there are no functions of the given type, write none on the line. 1. 2. 3. 4. 5. 6. Which of the above represent a partial function? _ ____________ Which of the above represent a total function? _ ___________ Which of the above represent a 1-1 function? _ ______________________ Which of the following represent an onto function? __ ___________________ Which of the above represent a 1-1 correspondence? _ ___________________ Which of the above are not functions? ___________________________________ Module 07 Short Answer Assignment - Functions, Domains, Ranges, 1-1, Onto - Answer Key For questions 7 - 18, your answers will be sets of numbers or letters associated with the points of the diagrams. Write your answers in set list form, enclosing each list of points in curly brackets. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. What is the domain of (A)? ________________________ What is the domain of definition of (A)? _________ What is the co-domain of (A)? ____________________________ What is the range (A)? ______________________________ What is the domain of (D)? ________________________________ What is the domain of definition of (D)? __________________________ What is the co-domain of (D)? _____________________________ What is the range of (D)? _______________________________________ What is the domain of (E)? _______________________________ What is the co-domain of (E)? ______________________________ What is the range of (E)? __________________________________ What is the domain of (H)? ________________________________ What is the co-domain of (H)? _____________________________ What is the range of (H)? __________________________________________ 2 Module 07 Written Assignment - Solutions 1 There are five exercises in this assignment, one per page. Use as much space as you like to answer the following exercises. Put your answers in the green-edged boxed provided. 1. Let X ={2,1, 0,1 } and Y ={2,1, 0,1 } . Define a function F : X Y follows: 2 F ( x )=x + x1 . Prove that F is neither 1 - 1 nor onto. as Module 07 Written Assignment - Solutions 2. For each of the following functions, either prove that the function is 1 - 1 or find a counterexample to show that the function is not 1 - 1. a. F:RR { 2 F ( x )= x 2for x 0 x for x 0 b. F:ZZ F ( n) = { n1 for n even n3 for n odd 2 Module 07 Written Assignment - Solutions 3 3. The first 3 parts of Theorem 2 are repeated below for your convenience. Also, recall that Theorem 2a was already proven in the Lesson. Prove Theorem 2, parts b and c. (Hint: once you have proven part b, you can use parts a and b to prove part c.) Theorem 2: Let F: X Y and G:Y Z be functions. Then a. If F and G are both 1 - 1 then G F is 1 - 1. b. If F and G are both onto then G F is onto. c. If F and G are both 1 - 1 correspondences then G F is a 1 - 1 correspondence. Let F:X Y and G:Y Z . If F and G are both onto then G F is onto. Proof: Let z Z. Since G is onto, there exists a y Y such that G(y) = z. Since F is onto, there exists an x X such that F(X) = y. Hence G F is onto G F( x ) = G( F( x)) = G(y) = z. So Module 07 Written Assignment - Solutions 4. Let X ={1, 2,3, 4, 5 } . Let three functions be defined as follows: F : X X with F ( 1 )=3, F ( 2 )=2, F ( 3 ) =2, F ( 4 )=2, F ( 5 )=5 G: X X with G ( 1 ) =1,G ( 2 )=3, G (3 )=4, G ( 4 )=5, G ( 5 ) =2 H: XX with H (1 ) =2, H ( 2 )=4, H ( 3 )=1, H ( 4 )=3, H ( 5 )=5 It is easier to do the problem if you write the functions in pair form: F={( 1, 3 ) , ( 2, 2 ) , ( 3, 2 ) , ( 4,2 ) , ( 5,5 ) } G={ (1, 1 ) , (2, 3 ) , ( 3, 4 ) , ( 4, 5 ) , (5, 2 ) } H={ (1, 2 ) , ( 2, 4 ) , ( 3,1 ) , ( 4, 3 ) , ( 5, 5 ) } Find each of the following. Give your answers as sets of ordered pairs. a. F G b. HF c. G H d. F G H e. F1 f. G g. H 1 h. Of 1 F 1 , 1 G , and H any) of these are functions? 1 , which (if 4 Module 07 Written Assignment - Solutions 5 5. Consider the following Boolean function F of two variables p and q defined by a table: p q 1 1 0 0 1 0 1 0 F(p,q ) 0 0 1 1 Is the combinatorial (logic) circuit shown below the correct circuit to represent this function? If not, describe how you would correct the circuit in order to make it accurately represent the function. Circuit p q F(p, q) p