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Relations, Functions, and Matrices
7. The function f: 5all English words6 S Z. In each case, find f (S ).
a. S = 5dog, cat, buffalo, giraffe6, f (x) = the number of characters in x
b. S = 5goose, geese, moose, Mississippi6, f (x) = the number of double-letter pairs in x
c. S = 5cheetah, seal, porpoise, koala6, f (x) = the number of e’s in x
8. The function f : 5binary strings6 S 5binary strings6. In each case, find f (S ).
a. S = 5000, 1011, 100016, f (x) = the second bit in x
b. S = 5111, 100, 01116, f (x) = the binary string that is the sum of the first and last bit
c. S = 5001, 11, 1016, f (x) = the binary string that is equal to x + 1
9. True or false:
a. An onto function means that every element in the codomain must have a unique preimage.
b. A one-to-one function means that every element in the codomain must have a unique preimage.
c. A one-to-one function means that no two elements in the domain map to the same element in the
codomain.
d. An onto function means that (the range) d (the codomain) = [.
10. True or false:
a. If every element in the domain has an image, it must be an onto function.
b. If every element in the codomain has an image, it must be an onto function.
c. If every element in the codomain has a preimage, it must be an onto function.
d. If the domain is the larger than the codomain, it can’t be a one-to-one function.
11. Let S = 50, 2, 4, 66 and T = 51, 3, 5, 76. Determine whether each of the following sets of ordered pairs
is a function with domain S and codomain T. If so, is it one-to-one? Is it onto?
a. 5(0, 2), (2, 4), (4, 6), (6, 0)6
b. 5(6, 3), (2, 1), (0, 3), (4, 5)6
c. 5(2, 3), (4, 7), (0, 1), (6, 5)6
d. 5(2, 1), (4, 5), (6, 3)6
e. 5(6, 1), (0, 3), (4, 1), (0, 7), (2, 5)6
12. For any bijections in Exercise 11, describe the inverse function.
13. Let S = the set of all U. S. citizens alive today. Which of the following are functions from domain S to the
codomain given? Which functions are one-to-one? Which functions are onto?
a. Codomain = the alphabet, f (person) = initial of person’s middle name
b. Codomain = the set of dates between January 1 and December 31, f (person) = person’s date of birth
c. Codomain = 9-digit numbers, f (person) = person’s Social Security number
14. Let S = the set of people at a meeting, let T = the set of all shoes in the room. Let f (x) = the left shoe x
is wearing.
a. Is this a function?
b. Is it one-to-one?
c. Is it onto?
Relations, Functions, and Matrices
7. The function f: 5all English words6 S Z. In each case, find f (S ).
a. S = 5dog, cat, buffalo, giraffe6, f (x) = the number of characters in x
b. S = 5goose, geese, moose, Mississippi6, f (x) = the number of double-letter pairs in x
c. S = 5cheetah, seal, porpoise, koala6, f (x) = the number of e’s in x
8. The function f : 5binary strings6 S 5binary strings6. In each case, find f (S ).
a. S = 5000, 1011, 100016, f (x) = the second bit in x
b. S = 5111, 100, 01116, f (x) = the binary string that is the sum of the first and last bit
c. S = 5001, 11, 1016, f (x) = the binary string that is equal to x + 1
9. True or false:
a. An onto function means that every element in the codomain must have a unique preimage.
b. A one-to-one function means that every element in the codomain must have a unique preimage.
c. A one-to-one function means that no two elements in the domain map to the same element in the
codomain.
d. An onto function means that (the range) d (the codomain) = [.
10. True or false:
a. If every element in the domain has an image, it must be an onto function.
b. If every element in the codomain has an image, it must be an onto function.
c. If every element in the codomain has a preimage, it must be an onto function.
d. If the domain is the larger than the codomain, it can’t be a one-to-one function.
11. Let S = 50, 2, 4, 66 and T = 51, 3, 5, 76. Determine whether each of the following sets of ordered pairs
is a function with domain S and codomain T. If so, is it one-to-one? Is it onto?
a. 5(0, 2), (2, 4), (4, 6), (6, 0)6
b. 5(6, 3), (2, 1), (0, 3), (4, 5)6
c. 5(2, 3), (4, 7), (0, 1), (6, 5)6
d. 5(2, 1), (4, 5), (6, 3)6
e. 5(6, 1), (0, 3), (4, 1), (0, 7), (2, 5)6
12. For any bijections in Exercise 11, describe the inverse function.
13. Let S = the set of all U. S. citizens alive today. Which of the following are functions from domain S to the
codomain given? Which functions are one-to-one? Which functions are onto?
a. Codomain = the alphabet, f (person) = initial of person’s middle name
b. Codomain = the set of dates between January 1 and December 31, f (person) = person’s date of birth
c. Codomain = 9-digit numbers, f (person) = person’s Social Security number
14. Let S = the set of people at a meeting, let T = the set of all shoes in the room. Let f (x) = the left shoe x
is wearing.
a. Is this a function?
b. Is it one-to-one?
c. Is it onto?