1. Use Theorem 1.1.1 to verify that (P Ù (~ (~P Ú Q) ) ) Ú (P Ù Q) Û P.
2. Use a truth table to determine if the following statements are logically equivalent.
P Ù ~Q ® R Û P Ù ~R ® Q.
3. Rewrite the statement in if-then form using quantifiers and variables.
“There are no easy questions on the exam.”
4. Write the following argument in logical form. Then determine if the argument is valid or invalid. If it is valid, state whether it is by Modus Ponens or Modus Tollens. If it is invalid, state whether it is an inverse or converse error.
Every adult is eligible to vote
John is eligible to vote
\ John is an adult
5. Use a truth table to determine if the following argument is valid or invalid.
P Ú Q
P ® ~Q
P ® R
\R
6. Design a circuit to take input signals P, Q, and R and output a 1 if, and only if all three input signals P, Q, and R have the same value.
7. Given the propositions:
P = The sun is shining
Q = I like to fish
R = I like to swim
And the statement: “If the sun is shining, then I like to fish or swim”
a) Write the statement in logical form using the propositions given.
b) Write the truth table for this statement.
c) Write the negation of the statement in logical form.
d) Write the contrapositive of the statement in logical form and in plain English.
8. Consider the following statement: Everybody trusts somebody.
a) Write the statement formally using the quantifiers " and $ and variables.
b) Write the negation of the statement in logical form
c) Write the negation of the statement in plain English.
9. A set of premises and conclusion are given. Use the Inference Rules to deduce the conclusion from the premises giving a reason for each step.
C ® D
B Ú D ® E
~E
\ ~C