Solutions to Problem Set 1

Math 310/520

9-1-2017

1. Write the negation (in words) of each statement.

(a) "57 is an odd integer."

(b) "$\pi$ is a rational number."

(c) "x is a negative real number."

(a) The negation is "57 is not an odd integer", or "57 is an even integer". (The negation is false, but you weren't asked about that.)

(b) The negation is "$\pi$ is not a rational number", or "$\pi$ is an irrational number".

(c) The negation is "x is a non-negative real number", or "x is greater than or equal to 0".

The negation is not "x is a positive number". 0 is not a negative number, but 0 is not a positive number.


2. Suppose

P is the statement "43 is prime".

Q is the statement "0 is an odd integer".

(a) Write the statement "43 is not prime or 0 is an odd integer" in symbols.

(b) Write the statement "0 is an even integer and 43 is not prime" in symbols.

(c) Write the statement $P \lor
   Q$ in words, and determine whether the statement is true or false.

(d) Write the statement $P \land
   Q$ in words, and determine whether the statement is true or false.

(a) $\lnot P \lor Q$ .

(b) $\lnot Q \land \lnot P$ .

(c) "43 is prime or 0 is an odd integer". The first part, "43 is prime" is true; hence, the "or" statement is true.

(d) "43 is prime and 0 is an odd integer". The first part, "43 is prime", is true. The second part, "0 is an odd integer", is false. (Actually, 0 is even.) Hence, the "and" statement is false.


3. Suppose that

F = "Phoebe likes french fries."

C = "Phoebe likes chili."

P = "Phoebe likes pizza."

(a) Translate the following logical statements into words:

(i) $\lnot P \ifthen \lnot C$ .

(ii) $(F \land \lnot C) \lor P$ .

(iii) $P \iff (C \land F)$ .

(b) Translate the following statements into logical symbols:

(i) "Phoebe likes pizza if Phoebe likes french fries."

(ii) "Phoebe likes chili and either Phoebe likes french fries or Phoebe likes pizza."

(iii) "If Phoebe likes french fries or Phoebe doesn't like pizza, then Phoebe likes chili."

(i) "If Phoebe doesn't like pizza, then Phoebe doesn't like chili."

(ii) "Either Phoebe likes french fries and Phoebe doesn't like chili or Phoebe likes pizza."

(iii) "Phoebe likes pizza if and only if both Phoebe likes chili and Phoebe likes french fries."

(b) (i) $F \ifthen P$ .

(ii) $C \land (F \lor P)$ .

(iii) $(F \lor \lnot P) \ifthen
   C$ .


Things do not get better by being left alone. - Winston Churchill


Contact information

Bruce Ikenaga's Home Page

Copyright 2017 by Bruce Ikenaga