Introduction to Discrete Math

 

Introduction to Discrete Math
The total number of points is 10. Your total score will be divided by 10 to produce a score over 100. Show all work unless
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1
Decide whether each of the following relations is a function:(3 points, 1 point each)

1. The domain and codomain are {1,2,3,4,5} and the relation R is given by the set of ordered pairs:

{(2,3),(3,3),(4,2),(5,1)}.
2. The domain and codomain are {1,2,3,4,5} and the relation R is given by the set of ordered pairs:

{(1,5),(2,3),(3,3),(1,2),(4,1)}.
3. The domain and codomain are the set of all people who were alive at midnight, December 31, 1999 and the relation R is
defined by the rule: {(x, y)|xand yaresiblings}.

2
Determine whether each function is one-to-one, onto, or both (check only one):(2 points, 1 point each)

1. g : Z × Zwhere g is defined by g(x) = x− 1
one-to-one: onto: both:
( x i f xiseven 1

2. f : N × Nwhere f is defined by f (x) =

2
x + 1 i f xisodd

one-to-one: onto: both:

3

Let P be the power set of {a,b,c}. A function f: P →Z, the set of integers, follows: For A in P, f(A)=the number of elements
in A.(2 points, 1 point each)

1. Is f one-to-one? Explain.

 
2. Is f onto? Explain.

4

1. List all the functions from the two-element set {1,2} to the three-element set {a,b,c}. (1 point)

2. Which functions, if any, are one-to-one?(1 point)

3. Which functions, if any, are onto?(1 point)
Q3
Let S={2,4,6,8} and T={1,5,7}.
What is the number of functions from S to T?
Q4
What is the name for a function that is one-to-one and onto?
Q5
What are permutation functions? Explain.