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.
1

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
Salads green
Main course Desserts salad fruit fish spaghetti pie cake milk
Beverages soda coffee x x x x x x x x x x
Paul x x 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.

a student S, such that

desserts D, S did not choose D. b.
 a student S, such that

salads T, S did not choose T. c.

desserts D,
 a student S such that S did not choose D. d.

beverages B,
 a student S such that S did not chose B. e.

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) by (4), (5), and basic algebra
(7)
 x, multiple of 2(x)

even(x) by definition of even numbers by (6) (8) multiple_of_2(S)

(9) even(S) 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)
(3) odd(Q)

(4)

p, integer(p) such that P = 2p+1

(5)

q, integer(q) such that Q = 2q+1 given in the problem given in the problem by (1), (2) and MP by (1), (3) and MP
(6) S = P + Q = 2p + 2q +2 = 2(p+q +1)
(7)
 x, multiple_of_2(x)

even(x) basic algebra 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) given in the problem

(3)

p, integer(p) such that P = 2p by (1), (2) and MP
(4)
 x, odd(x)
 
p, integer(p) & x = 2p+1 by definition
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(5) odd(Q)

(6)

q, integer(q) such that Q = 2q+1
(7) S = P + Q = 2p + 2q +1 = 2(p+q) +1 given in the problem by (4), (5) and MP basic algebra
(8) The sum of two integers is an integer basic algebra
(9)
 x, x = 2k+1

odd(x) by definition by (7) (10) S = 2k + 1, where k = p+q

(11) odd(S) by (9), (10) and MP
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