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1. Let p and...

1. Let p and q be propositions

p : you drive over 65 miles per hour.

q : you get a speeding a speeding ticket.

Write these propositions using p and q and logical connectives

a) You drive over 65 miles per hour, but you do not get a speeding ticket.

b) If you do not get a speeding ticket then you are not driving over 65 miles per hour.

2. Construct a truth table for the following logical expression, (p q) è (p q).

3. Show that (p è q) and ~q è ~p are logically equivalent using truth tables.

4. Are the following statements logically equivalent?

p q r ; p q r) ;

Page 2

5. Let the universe be defined as U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} with the following

subsets: A = {2, 4, 6, 8}, B = {1, 3, 6, 8, 10}, C = {2, 3, 4, 5, 7, 8, 9}

Find the following sets:

a) A ∩ C

b) B ∩ (~C) where ~C is the complement of set C relative to the universe, U.

6. Let the universe be defined as U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} with the following

subsets: A = {3, 4, 5, 8}, B = {1, 3, 6, 8, 10}, C = {2, 3, 4, 5, 7, 8, 9}. A set can be

stored in a computer as a string of 1s or 0s. Set U can be represented as the following

string 11 1111 1111. How can we represent each of the subsets A, B and C as strings

of length ten?

7. How many strings of three decimal digits (0, 1, 2, …, 9)

a) do not contain the same digit three times?

b) begin with an odd digit?

8. There are four major auto routes from Boston to Detroit and six from Detroit to Los

Angeles. How many major auto routes are there from Boston to Los Angeles via

Detroit?

9. How many strings are there of lowercase letters of length four or less?

Page 3

10. How many orders can five runners finish a race if no ties are allowed?

11. How many permutations of the letters ABCDEFGH contain the string CDE?

p : you drive over 65 miles per hour.

q : you get a speeding a speeding ticket.

Write these propositions using p and q and logical connectives

a) You drive over 65 miles per hour, but you do not get a speeding ticket.

b) If you do not get a speeding ticket then you are not driving over 65 miles per hour.

2. Construct a truth table for the following logical expression, (p q) è (p q).

3. Show that (p è q) and ~q è ~p are logically equivalent using truth tables.

4. Are the following statements logically equivalent?

p q r ; p q r) ;

Page 2

5. Let the universe be defined as U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} with the following

subsets: A = {2, 4, 6, 8}, B = {1, 3, 6, 8, 10}, C = {2, 3, 4, 5, 7, 8, 9}

Find the following sets:

a) A ∩ C

b) B ∩ (~C) where ~C is the complement of set C relative to the universe, U.

6. Let the universe be defined as U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} with the following

subsets: A = {3, 4, 5, 8}, B = {1, 3, 6, 8, 10}, C = {2, 3, 4, 5, 7, 8, 9}. A set can be

stored in a computer as a string of 1s or 0s. Set U can be represented as the following

string 11 1111 1111. How can we represent each of the subsets A, B and C as strings

of length ten?

7. How many strings of three decimal digits (0, 1, 2, …, 9)

a) do not contain the same digit three times?

b) begin with an odd digit?

8. There are four major auto routes from Boston to Detroit and six from Detroit to Los

Angeles. How many major auto routes are there from Boston to Los Angeles via

Detroit?

9. How many strings are there of lowercase letters of length four or less?

Page 3

10. How many orders can five runners finish a race if no ties are allowed?

11. How many permutations of the letters ABCDEFGH contain the string CDE?

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