Download HW-06 due 02/22

CmSc180 Discrete Mathematics
Homework 06 due 02/22
1. Determine whether the following arguments are valid or invalid.
a.
Sandra knows COBOL and Sandra knows C.
Therefore Sandra knows C
b.
If the number is larger than 2, then its square is larger than 4.
This number is not larger than 2.
Therefore the square of this number is not larger than 4.
c.
If at least one of these two numbers is divisible by 6, then the product is
divisible by 3.
Neither of these two numbers is divisible by 6.
Therefore the product is not divisible by 3.
d.
If there are as many rational numbers as there are irrational numbers, then
the set of all rational numbers is infinite.
The set of all rational numbers is infinite.
Therefore there are as many rational numbers as there are irrational
numbers.
2. Suppose you are visiting an island where two kinds of people live: knights who
always tell the truth, and knaves who always lie. Two natives A and B approach
you, and B says: "Both of us are knaves".
What are A and B?
Explain your reasoning.
3. Prove by mathematical induction
1 + 2 + 22 + 23 + … + 2n = 2(n+1) - 1, n = 0, 1, 2, 3, …
4. Let the universal set be the set R of all real numbers and let
A = {x  R | -2  x  3}, B = { x  R | -1 < x < 4}.
Find each of the following:
AB
=
AB
=
~A
=
1
~B
=
~A  ~B
=
~A  ~B
=
~(A  B)
=
~(A  B)
=
5. Prove using element proof that
(A - B) - C = A - (B  C)
6. Find the powerset of
6.1. A = {{a}, {b}}
6.2. A = 
6.3. A = {}
7. Let the universal set be all people. Within the universal set, A is the set of all
computer programmers, B is the set of all accountants, C is the set of all women,
D is the set of all people of age 40 and older. What are the following sets:
a. ~(A  B)
b. (A  B)  ~C
c. ~D  C  A
8. Let A = {1,2,3,4,5}, B = {a, b, c, d, e}
a. Give a one-to-one function from A to B. Is it a bijection?
b. Give a function from A to B which is not one-to-one. Is it an onto function?
c. Give a function from A to B which is not onto. Is it one-to-one?
9. Let A = {a,b,c}, B = {1,2}
a. Give an onto function from A to B. Is it one-to-one?
b. Give a function from B to A which is one-to-one. Is it “onto”?
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