The following problems are based on problems in Johnsonbaugh, Discrete Mathematics. 1997: Prentice-Hall.
1. Tell whether the following are: Propositions; Propositional Functions; or Neither. For each proposition, determine its truth or falsity; for each propositional function, specify a possible universe of discourse.
a. Star Wars won the Academy Award for best picture of 1947.
b. X > 5
c. Let y denote a real number greater than 10.
d. The train goes more than 60 mph.
e. Take my wife, please.
f. (2p + 1) is an odd integer.
2. Let W(x, y) denote the propositional function "x weighs more than
y," with the condition that x ¹ y,
and the universe of discourse be the following five people:
Alan
(150 lbs.)
Bill
(185 lbs.)
Cheryl (140 lbs.)
Doug (220
lbs.)
Elaine
(120 lbs.)
Write each of the following in words and tell whether it is true or
false.
a. "x "y: W(x,
y)
b. $x "y: W(
x, y)
c. $ x $y: W(
x, y)
3. For the following problems, the universe of discourse is the set
of all integers. Write each of the following symbolically and tell whether
it is true or false.
a. For all x, x2 > x
b. For some x, if x > 1, then x - 1 > 0
c. For all x, for all y, x2 > y + 1
d. For all x, for some y, x2 > y + 1
e. For all x, for all y, x2 + y2 > 9
f. For some x, for all y, x * y = x.
g. For all x, for some y, x * y = x.
h. For all x, for all y, x2 + y2 ³
0.
4. Johnsonbaugh quotes economist Robert J. Samuelson as saying, "Every environmental problem is not a tragedy." This is an example of an ambiguity, a statement which can be interpreted in more than way. A frequent challenge in programming is to find and resolve all ambiguities in a problem specification. Find both ways of writing this statement symbolically; decide which one you think Samuelson meant; and write that meaning (in English) in a non-ambiguous way.
5. For each of the following, specify the universe of discourse; propositions
P(x), Q(x), etc.; and write the statement symbolically.
a. Every cloud has a silver lining.
b. There are no atheists in foxholes.
c. You can lead a horse to water but you can't make it drink.
d. All mathematicians are eccentric.
e. All students who cheat on tests will be expelled.
f. Somebody's been eating my porridge.
This page last modified 9/5/1999.