All the following statements are false or incorrect?
Instructions
- Show your work and justify all your answers for the following problems to receive full credit.
- You may work directly in this exam packet (print it, scan it, submit it) or present your work neatly
and labeled on separate paper.
- Make sure your symbols for set operations and logical connectives are clear. If you prefer, you may
use explicit words like “NOT”, “OR”, “COMPLEMENT”, “UNION”, etc. instead of special characters.
- Answers that are correct but unexplained or not substantiated by work, or answers that are copied
from elsewhere, will be marked a zero. If this recurs, the entire assignment will be marked a zero.
Problem 1 (28 pts)
All the following statements are false or incorrect. Explain why. Provide a direct counterexample or a simple Venn diagram, if necessary, to make your point.
- In base 12 (with duodecimal “digits” A = 10,B = 11), the following sum holds:
1A + 2B = 46
- If you write the cube of an even decimal number in binary form, it always ends in a 1.
- The intersection of two disjoint sets has no subsets.
- The negation of the statement “Apples are red” is “Apples are not red.”
- The following argument is valid:
She doesn’t like my lasagna when I put sausages in it. She didn’t like my
lasagna yesterday. So I must have put sausages in it.”
- If a pizzeria offers 5 different toppings to put on their pizzas (assume it also prepares
pizzas with no toppings), then it offers a total of exactly 25 pizzas with at least 2 toppings.
- The compound statement (p⋁~v) ↔ r is never true if the components p and r are false.
Problem 2 (6 pts)
Let N be the set of distinct letters in the word “Nassau” and S be the set of distinct letters in
the word “Suffolk.”
a. Find the set (N ∪ S) − (N ∩ S).
b. List all the subsets of N.
Problem 3 (6 pts)
Consider the following conditional statement: “Peanuts are neither peas nor nuts.”
a) Write the contrapositive of this statement.
b) Write the negation of this statement.
Problem 4 (8 pts)
A recent culinary survey asked 50 chefs on Long Island whether they are currently cooking French
and/or Italian dishes at their restaurants. Here are the results of that survey:
• 16 chefs said they feature French dishes on their menus.
• 22 chefs said they feature Italian dishes on their menus.
• 21 chefs said they feature neither French nor Italian dishes on their menus.
Draw a Venn diagram for this survey below.
a. How many chefs feature only French dishes on their menus? __
b. How many chefs feature both Italian and French dishes on their menu? __
c. How many chefs do not feature Italian dishes on their menu? ______
Problem 5 (4 pts)
Show that the statement (p ∧ q) → (p ∨ q) is a tautology without using a truth table.
Problem 6 (10 pts)
Your friend tells you the following:
“If I get a puppy this summer, then my sister will not come to my pool party. My sister
will come to my pool party or my mother will be upset. I ultimately decided not to get a
puppy this summer. So now I know my mother won’t be upset!”
a. Show your friend’s argument in symbolic form using three basic statements p, q and r. Write
out these components explicitly.
b. Analyze the argument by any means (truth values, truth table, standard forms). Is your friend’s
logic valid or not
