Download HW-04: Solutions - Simpson College

CmSc180 Discrete Mathematics
Homework 04 Solutions
1. "If compound X is boiling then its temperature must be at least 250°F"
Assuming that this statement is true, which of the following statements must also
be true:
a. If the temperature of compound X is at least 250°F, then compound X is boiling
b. If the temperature of compound X is less than 250°F, then compound X is not
boiling
c. Compound X will boil only if its temperature is at least 250°F.
d. If compound X is not boiling then its temperature is less than 250°F.
Explain your answer. This means to show that your answer is correct using the
properties of the implication P  Q
Answer: The sentence "If compound X is boiling then its temperature must be at
least 250°F" can be represented as P  Q, where P = “compound X is boiling” and Q =
“its temperature must be at least 250°F”.
Consequently, for the sentences a), b), c), and d) we have:
a) Q  P , this is the converse of P  Q
b) ~Q  ~P, this is the contrapositive of P  Q
c) P only if Q, equivalent to P  Q
d) ~P  ~Q, this is the inverse of P  Q
The true statements are b) and c) because the contrapositive is equivalent to the
implication, and P only if Q is equivalent to P Q
2. A college cafeteria line has four stations: salads, main courses, desserts, and
beverages.
The salad station offers a choice of green salad or fruit salad;
the main course station offers spaghetti or fish;
the dessert station offers pie or cake;
and the beverage station offers milk, soda, or coffee.
Three students Ann, Paul, and Tim, go through the line and make the following
choices:
Ann: green salad, spaghetti, pie, milk
Tim: fruit salad, fish, pie, cake, milk, and coffee
Paul: spaghetti, fish, pie, soda.
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A. Determine whether each of the following statements is true or false:
a.  students S,  a dessert D such that S chose D.
b.  students S,  a salad T such that S chose T.
c.  a dessert D such that  students S, S chose D.
d.  a beverage B such that  students S, S chose B.
e.  an item I such that  students S, S did not choose I
B. Write the negations of the above statements, using the same format of
representation.
Example: The negation of statement (a) would be:
 a student S, such that  desserts D, S did not choose D
Student
Ann
Tim
Paul
Salads
green
salad
fruit
x
x
Main course
fish
x
x
spaghetti
x
x
Desserts
pie
x
x
x
cake
x
Beverages
milk
x
x
soda
coffee
x
x
A. Determine whether each of the following statements is true or false:
True statements are in boldface:
f.  students S,  a dessert D such that S chose D.
g.  students S,  a salad T such that S chose T.
h.  a dessert D such that  students S, S chose D.
i.  a beverage B such that  students S, S chose B.
j.  an item I such that  students S, S did not choose I
B. Write the negations of the above statements, using the same format of
representation.
Example: The negation of statement (a) would be:
 a student S, such that  desserts D, S did not choose D
a.
b.
c.
d.
e.
 a student S, such that  desserts D, S did not choose D.
a student S, such that  salads T, S did not choose T.
 desserts D, a student S such that S did not choose D.
 beverages B, a student S such that S did not chose B.
 items I, a student S such that S chose I
3. Give direct proof for the following statements
3.a. The sum of two odd numbers is even
3.b. The sum of an even and an odd number is odd
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Example:
Prove that sum of two even numbers is even.
Proof:
Let P and Q be two even numbers
(1) x, even(x)  multiple of 2(x), i.e.  p, integer(p) & x = 2p
(2) even(P)
given in the problem
(3) even(Q)
given in the problem
 (4)  p, integer(p) such that P = 2p by (1), (2) and MP
 (5)  q, integer(q) such that Q = 2q by (1), (3) and MP
(6) S = P + Q = 2p + 2q = 2(p+q)
(7) x, multiple of 2(x)  even(x)
(8) multiple_of_2(S)
 (9) even(S)
by (4), (5), and basic algebra
by definition of even numbers
by (6)
by (7), (8) and MP
Solution
3.a. The sum of two odd numbers is even
Let P and Q be two odd numbers
(1) x, odd(x)   p, integer(p) & x = 2p+1
(2) odd(P)
given in the problem
(3) odd(Q)
given in the problem
 (4)  p, integer(p) such that P = 2p+1
by (1), (2) and MP
 (5)  q, integer(q) such that Q = 2q+1
by (1), (3) and MP
(6) S = P + Q = 2p + 2q +2 = 2(p+q +1)
basic algebra
(7) x, multiple_of_2(x)  even(x)
by definition
(8) multiple_of_2(S)
by (6)
 (9) even(S)
by (7), (8) and MP
3.b. The sum of an even and an odd number is odd
Let P be an even number, and Q be an odd numbers
(1) x, even(x)  multiple of 2(x), i.e.  p, integer(p) & x = 2p
(2) even(P)
 (3)  p, integer(p) such that P = 2p
(4) x, odd(x)   p, integer(p) & x = 2p+1
given in the problem
by (1), (2) and MP
by definition
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(5) odd(Q)
 (6)  q, integer(q) such that Q = 2q+1
given in the problem
by (4), (5) and MP
(7) S = P + Q = 2p + 2q +1 = 2(p+q) +1
basic algebra
(8) The sum of two integers is an integer
basic algebra
(9) x, x = 2k+1  odd(x)
by definition
(10) S = 2k + 1, where k = p+q
by (7)
 (11) odd(S)
by (9), (10) and MP
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