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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