M230 Exams

Repeat problem 2, except with Poison instead of Takeaway.

𝑃(𝑛) = "𝐴 𝑝𝑙𝑎𝑦𝑒𝑟 𝑤ℎ𝑜 𝑙𝑒𝑎𝑣𝑒𝑠 (𝑘 +1)𝑛 +1 𝑐𝑜𝑖𝑛𝑠 𝑚𝑎𝑦 𝑓𝑜𝑟𝑐𝑒 𝑎 𝑤𝑖𝑛."

Let 𝑘 ∈ 𝒩 and consider the game of Takeaway in which each player may take anywhere from 1 to k coins on their turn. Write a propositional function similar to the one in problem 1 that expresses the winning strategy to this game.

𝑃(𝑛) = "𝐴 𝑝𝑙𝑎𝑦𝑒𝑟 𝑤ℎ𝑜 𝑙𝑒𝑎𝑣𝑒𝑠 (𝑘 +1)𝑛 𝑐𝑜𝑖𝑛𝑠 𝑚𝑎𝑦 𝑓𝑜𝑟𝑐𝑒 𝑎 𝑤𝑖𝑛."

Let A and B be sets. Which of the following is logically equivalent to 𝐴 ⊂ 𝐵? a. 𝐴 ⊆ 𝐵 𝑜𝑟 𝐵 ⊆ 𝐴 b. 𝑨 ⊆ 𝑩 & 𝑨 ≠ 𝑩 c. ∀𝑥 ∈ 𝐴,𝑥 ∉ 𝐵 d. |𝐴| < |𝐵|

𝑨 ⊆ 𝑩 & 𝑨 ≠ 𝑩

Prove that the following argument is valid or write a counter-example: All cows are mammals. Some cows have horns. ∴ Some mammals have horns. Valid C(x)= "x is a cow" H(x)= "x has horns" M(x)= "x is a mammal"

1) Suppose ∀𝑥(𝐶(𝑥) → 𝑀(𝑥)) 2) Suppose ∃𝑥(𝐶(𝑥)&𝐻(𝑥)) 3) Show: ∃𝑥(𝑀(𝑥)&𝐻(𝑥)) 4) From line 2, we can let a be such that 𝐶(𝑎)&𝐻(𝑎) is true. 5) From line 1, 𝐶(𝑎) → 𝑀(𝑎). 6) Since we also know that 𝐶(𝑎) is true from line 4, we can conclude 𝑀(𝑎) by modus ponens. 7) Thus, 𝐻(𝑎)&𝑀(𝑎) are both true from lines 4 and 6. 8) ∴ Some mammals have horns, as a is a mammal with horns. Q.E.D.

Prove that the following argument is valid: P & Q P → R (P & R) → S ∴Q & S

1) Suppose 𝑃&𝑄 2) Suppose 𝑃 → 𝑅 3) Suppose (𝑃&𝑅) → 𝑆 4) Show: 𝑄&𝑆 5) From line 1, we know P is true. 6) Since 𝑃 → 𝑅 (2), we know 𝑅 is also true. 7) Thus 𝑃&𝑅 are both true 8) But (𝑃&𝑅) → 𝑆 (3), so 𝑆 must be true as well. 9) ∴ 𝑄&𝑆 are both true from lines 1 and 8. Q.E.D.

Consider the following argument: No Montagues are Capulets. Romeo is a Montague. Therefore, Romeo is not a Capulet. a. Let M(x)= "x is a Montagues," C(x)= "x is a Capulet" r= Romeo and symbolize the argument above. b. Symbolize the same argument, except instead of using propositional functions, define relevant sets. Let M be the set of Montagues and C be the set of Capulet's

A. Many choices for the first one: ~∃𝑥(𝑀(𝑥)&𝐶(𝑥)),∀𝑥(~𝑀(𝑥) 𝑜𝑟 ~𝐶(𝑥)),𝑒𝑡𝑐. 𝑀(𝑟) ∴ ~𝐶(𝑟) B. 𝑀 ∩ 𝐶 = ∅; ∀𝑥 ∈ 𝑀,𝑥 ∉ 𝐶;∀𝑥(𝑥 ∈ 𝑀 → 𝑥 ∉ 𝐶);𝑒𝑡𝑐. 𝑟 ∈ 𝑀 ∴ 𝑟 ∉ 𝐶

Symbolize each of the following statements related to numbers: a. Every natural number is greater than or equal to 1. b. Every integer is a real number. c. Some real numbers are not rational. d. Every prime number except for 2 is an odd number. (Let P(x)= "x is prime" O(x)= "x is odd", use the set of natural number as your universe for this one) ∀𝑥 ∈ 𝒩 − {2},𝑃(𝑥) → 𝑂(𝑥) ∀𝑥 ∈ 𝒩,𝑥 ≠ 2 → (𝑃(𝑥) → 𝑂(𝑥))

A. ∀𝑥 ∈ 𝒩,𝑥 ≥ 1 B. ∀𝑥 ∈ 𝒵,𝑥 ∈ ℛ C. (Let Q(x)= "x is rational") ∃𝑥 ∈ ℛ,~𝑄(𝑥) D. (Let P(x)= "x is prime" O(x)= "x is odd", use the set of natural number as your universe for this one) ∀𝑥 ∈ 𝒩 − {2},𝑃(𝑥) → 𝑂(𝑥) ∀𝑥 ∈ 𝒩,𝑥 ≠ 2 → (𝑃(𝑥) → 𝑂(𝑥))

The principle of mathematical induction tells us that: [∀𝑛 ∈ 𝒩,𝑃(𝑛)] ↔ [𝑃(1)& ∀𝑛 ∈ 𝒩,𝑃(𝑛) → 𝑃(𝑛 +1)] A. Use the negation rule and DeMorgan's Law to write a negation of: [𝑃(1)& ∀𝑛 ∈ 𝒩,𝑃(𝑛) → 𝑃(𝑛 +1)] B.Let 𝑃(𝑛) = "I can lift a bag that weighs n milligrams." Write a sentence in the form of your answer to (a) to express that ∀𝑛 ∈ 𝒩,𝑃(𝑛) is false.

A.~𝑃(1)∨∃𝑛 ∈ 𝒩 𝑠.𝑡.[𝑃(𝑛)&~𝑃(𝑛 +1)] B.Either, I can't lift a bag that is 1 milligram, or there exists some natural number n for which I can lift a bag that weighs n milligrams, but cannot lift a bag that weighs n+1 milligrams

Prove that the following argument is valid or show a counter-example: All turtles are reptiles. No turtles are warm blooded. ∴ No Reptiles are warm blooded.

Invalid Set of turtles = {𝑎}, Set of reptiles = {𝑎,𝑏}, Set of warm blooded things = {𝑏} Thus, all turtles are reptiles and no turtles are warm blooded, but the conclusion that no reptiles are warm blooded is false

Prove by PMI that ∀𝑛 ∈ 𝒩,∑ 2𝑖 = n(n+1)

Let P(n)= ∑ 2𝑖 𝑛 𝑖=1 = 𝑛(𝑛 +1) Base case: Show P(1) ∑ 2𝑖 1 𝑖=1 = 2 = 1(1+1), so P(1) is true. Inductive Step: Show ∀𝑛 ∈ 𝒩,𝑃(𝑛) → 𝑃(𝑛 +1) Assume 𝑘 ∈ 𝒩 𝑠.𝑡.𝑃(𝑘) is true, that is ∑ 2𝑖 𝑘 𝑖=1 = 𝑘(𝑘 +1) We want to show that 𝑃(𝑘 +1) follows. We can add 2(k+1) to both sides to obtain ∑ 2𝑖 𝑘 𝑖=1 +2(𝑘 +1) = 𝑘(𝑘 +1)+2(𝑘 +1) ∴ ∑ 2𝑖 𝑘+1 𝑖=1 = 𝑘2 +𝑘 +2𝑘 +2 = 𝑘2 +3𝑘 +2 ∴ ∑ 2𝑖 𝑘+1 𝑖=1 = (𝑘 +1)(𝑘 +2) The above line is simply 𝑃(𝑘 +1). Thus, we have shown that the truth of P(k) always implies the truth of P(k+1). Thus ∀𝑛 ∈ 𝒩,𝑃(𝑛) → 𝑃(𝑛 +1) Thus, by PMI, we conclude that ∀𝑛 ∈ 𝒩,𝑃(𝑛). Q.E.D.

Consider the game of Takeaway in which each player may take either 1 or 2 coins on their turn. Use the Principle of Mathematical Induction to prove the following statement is true for all values of n: "A player who leaves 3n coins may force a win."

Let 𝑃(𝑛) = "𝐴 𝑝𝑙𝑎𝑦𝑒𝑟 𝑤ℎ𝑜 𝑙𝑒𝑎𝑣𝑒𝑠 3𝑛 𝑐𝑜𝑖𝑛𝑠 𝑚𝑎𝑦 𝑓𝑜𝑟𝑐𝑒 𝑎 𝑤𝑖𝑛." Show: 𝑃(0), that is "A player who leaves 0 coins may force a win." The rules state that a player who leaves 0 coins, that is takes the last coin, wins. Show: ∀𝑛 ∈ 𝒩0,𝑃(𝑛) → 𝑃(𝑛 +1) Assume 𝑘 ∈ 𝒩0 has the property that 𝑃(𝑘) is true, that is a player who leaves 3k coins may force a win. Now suppose that a player leaves 3(𝑘 +1) coins. If their opponent takes 1 coin, they can respond by taking 2 coins. If their opponent takes 2 coins, they can respond by taking 1 coin. In either case, they have left 3(𝑘 +1)−3 = 3𝑘 +3−3 = 3𝑘 coins. From this point, our assumption tells us that the player can force a win. Thus, for any value k, if we assume 𝑃(𝑘), we see that 𝑃(𝑘+1) follows, so ∀𝑛 ∈ 𝒩0,𝑃(𝑛) → 𝑃(𝑛 +1) ∴ By the Principle of mathematical induction we have ∀𝑛 ∈ 𝒩0,𝑃(𝑛). Q.E.D.

Symbolize the following. You may use sets or propositions (or mix and match as you like), but make sure to define all of your terms: Any shape is a square if and only if it both has four equal sides and four equal angles.

Lots of ways to do this one. One example: Let S be the set of shapes, Q(x)= "x is a square" E(x)= "x has four equal sides" A(x)= "x has four equal angles" ∀𝑥 ∈ 𝑆(𝑄(𝑥) ↔ (𝐸(𝑥)&𝐴(𝑥))

Consider the game of Takeaway in which players can take either 1, 3, 4 or 5 coins on each turn. Find the general solution to this game (you don't need to prove it). If the game begins with 282 coins, which player can force a win?

P(n)= "A player can force a win if they can leave 8n or 8n+2 coins." Divide 282 by 8 to obtain 35 remainder 2. So 282 = 35*8 +2. Thus, the second player can force a win. Since the game starts with a number of the form 8n+2, it's as if the second player left that many coins for the first player and so they can force a win.

There are three gladiators with strengths of 3, 4 and 5. When two gladiators fight, the weaker one is killed, while the stronger one's strength is reduced by the value of the weaker one (two gladiators of the same strength will kill each other simultaneously). In this scenario, the strongest gladiator will get to choose their first opponent. After the first fight, the winner will face the remaining gladiator. Prove that the gladiator of strength 5 is a kingmaker.

The gladiator has two choices: Choice 1: They can fight the gladiator with 4 strength. They will win this fight, but be reduced to 1 strength. Then they will be defeated by the gladiator with 3 strength. Choice 2: They can fight the gladiator with 3 strength. They will win this fight, but be reduced to 2 strength. Then they will be defeated by the gladiator with 4 strength. Thus, no matter what choice the gladiator with 5 strength makes, they will lose. However, the choice that they make determines which other gladiator does win.

It is currently the yellow player's turn to place a chip into the Connect 4 Board. Prove that the yellow player can force a win from the position shown below. Notes: For notation, you can call the columns, "column 1," "column 2," etc. where column 1 is the column on the far left and the numbers go from left to right. Also, in a game of connect 4, the chip may be placed in any column with an empty spot and will always fall to the lowest empty spot in a column. A player wins by having four consecutive chips of their color, whether those 4 chips form a row, a column or a diagonal.

The yellow player may force a win as follows: First they place their chip in column 5. Case 1: The red player does not respond by placing in column 5. In none of these cases does the red player get 4 red chips. This allows the yellow player to play in column 5 to connect a row of 5 yellow chips. Case 2: The red player does respond in column 5 Again, the yellow player responds by also playing in column 5. In this case, they connect 4 diagonally. Thus, the yellow player has a winning move available regardless what the red player does and thus can force a win. Q.E.D.

Suppose that you have a propositional function P defined on 𝒩0. You are able to prove two things about P: i) 𝑃(0) is true. Ii) ∀𝑛 ∈ 𝒩0,𝑃(𝑛) → 𝑃(𝑛 +3). For which numbers can you be certain that 𝑃 is true?

We can be certain that P is true for every multiple of 3.

Which of the following is not a tautology: a. (A -->B) <--> (B-->A) b. (A OR B) <--> (B OR A) c. (A AND B) <--> (B AND A) d. (A <-->B) <--> (B<-->A)

a. (A -->B) <--> (B-->A)

On which line do you conclude ~𝑃? 1) Suppose 𝑃 → 𝑄 2) Suppose 𝑄 ∨ 𝑅 3) Suppose (𝑅 & ~𝑃) → 𝑆 4) Suppose ~𝑄 5) Show: S 6) From (1), (4) and modus tollens, we conclude @@ 7) From (2), (4) and wedge-out, we conclude ## 8) So from (6) and (7) we have %% 9) Thus, we can conclude ** from (3), (8) and modus ponens. a. 6 b. 7 c. 8 d. 9

a. 6

Which of the following is not a tautology? a. A OR ~B b. A OR ~A c. A <--> A d. A OR (A--> B)

a. A OR ~B

Suppose you want to "Show: (𝐴&𝐵) → (𝐶 ↔ 𝐷)." What should be the first line that you write after the show line, using our conditional derivation. a. Assume 𝑨&𝑩 b. Assume 𝐶 ↔ 𝐷 c. You either assume A or you assume B, but you don't assume both. d. You don't assume anything. Just work with the premises given

a. Assume 𝑨&𝑩

All squares are rectangles. All rectangles have four sides. /All squares have four sides. This is: a. Factually correct and valid. b. Factually correct but not valid. c. Valid, but not factually correct. d. Neither factually correct nor valid.

a. Factually correct and valid.

The statement "all triangles have three sides" is equivalent to which of the following always being true: a. Given any shape, if a shape is a triangle, then it has three sides. b. Given any shape, if a shape has three sides then it is a triangle. c. There is a shape that is both a triangle and has three sides. d. None of the above.

a. Given any shape, if a shape is a triangle, then it has three sides.

1. If the Dodgers win at least 5 more games, they will win the NL West. If they win the NL West, they will make the playoffs. You can conclude: a. If the Dodgers win at least 5 more games, they will make the playoffs. b. If the Dodgers make the playoffs, they will win the NL West. c. If the Dodgers win the NL West, they will win at least 5 more games. d. All of the above.

a. If the Dodgers win at least 5 more games, they will make the playoffs.

1. Jane has a dog or Jane has a cat. If Jane has a cat, then Jane has a litterbox. Jane does not have a dog. You can conclude: a. Jane has a litterbox. b. Jane does not have a litterbox. c. There is insufficient information to determine whether Jane has a litterbox. d. The premises are contradictory, so any conclusion is meaningless.

a. Jane has a litterbox.

1. All crickets are insects. If Jiminy is an insect, then Jiminy has six legs. Jiminy is a cricket. a. Jiminy must have six legs b. Jiminy must not have six legs c. There is insufficient information to determine whether or not Jiminy has six legs. d. The premises are contradictory, so any conclusion is meaningless.

a. Jiminy must have six legs

If Carlos is at least 5 feet tall or Carlos is at least 16 years old, then Carlos can ride the roller coaster. Carlos is 18 years old. Carlos cannot ride the roller coaster. You can conclude: a. The premises are contradictory so any conclusion is meaningless. b. Carlos is at least 5 feet tall c. Carlos is under 5 feet tall d. There is insufficient information to conclude anything about Carlos' height.

a. The premises are contradictory so any conclusion is meaningless.

Let S be the set of Mr. Boone's students. 𝑊(𝑥)= "x will do well on the exam," 𝐻(𝑥)= "x will be happy," b= Mr. Boone. Symbolize: "If all of Mr. Boone's students do well on the exam, then Mr. Boone will be happy." a. [∀𝒙 ∈ 𝑺,𝑾(𝒙)] → 𝑯(𝒃) b. ∀𝑥 ∈ 𝑆,(𝑊(𝑥) → 𝐻(𝑏)) c. ∀𝑥 ∈ 𝑆,(𝑊(𝑥) → 𝐻(𝑥)) d. ∀𝑥 ∈ 𝑆,(𝐻(𝑏) → 𝑊(𝑥)

a. [∀𝒙 ∈ 𝑺,𝑾(𝒙)] → 𝑯(𝒃)

Which of the following is a negation of ∃𝑥 ∈ 𝑃∀𝑦 ∈ 𝑄(𝐴(𝑥,𝑦)&𝐵(𝑥,𝑦))? a. ∀𝒙 ∈ 𝑷∃𝒚 ∈ 𝑸(~𝑨(𝒙,𝒚) ∨ ~𝑩(𝒙,𝒚)) b. ∀𝑥 ∈ 𝑃∃𝑦 ∈ 𝑄(~𝐴(𝑥,𝑦)& ~𝐵(𝑥,𝑦)) c. ∃𝑥 ∈ 𝑃∀𝑦 ∈ 𝑄(~𝐴(𝑥,𝑦) 𝑉 ~𝐵(𝑥,𝑦)) d. ∀𝑦 ∈ 𝑄∃𝑥 ∈ 𝑃(𝐴(𝑥,𝑦) & 𝐵(𝑥,𝑦))

a. ∀𝒙 ∈ 𝑷∃𝒚 ∈ 𝑸(~𝑨(𝒙,𝒚) ∨ ~𝑩(𝒙,𝒚))

Let B(x) = "x is a bird," M(x)= "x is a mammal," E(x)= "x lays eggs." Symbolize: All birds and some mammals lay eggs. a. ∀𝒙(𝑩(𝒙) → 𝑬(𝒙)) & ∃𝒙(𝑴(𝒙) & 𝑬(𝒙)) b. ∀𝑥((𝐵(𝑥) → 𝐸(𝑥))& (𝑀(𝑥)&𝐸(𝑥))) c. ∀𝑥(𝐵(𝑥) → 𝐸(𝑥)) & ∃𝑥(𝑀(𝑥) → 𝐸(𝑥)) d. ∀𝑥((𝐵(𝑥)&𝑀(𝑥)) → 𝐸(𝑥))

a. ∀𝒙(𝑩(𝒙) → 𝑬(𝒙)) & ∃𝒙(𝑴(𝒙) & 𝑬(𝒙))

Which of the following statements is false? a. ∃𝒙 ∈ 𝓩 𝒔.𝒕.𝒙𝟐 = 𝟖 b. ∃𝑥 ∈ ℛ 𝑠.𝑡.𝑥2 = 8 c. ∃𝑥 ∈ 𝒵 𝑠.𝑡.𝑥2 = 9 d. ∃𝑥 ∈ ℛ 𝑠.𝑡.𝑥2 = 9

a. ∃𝒙 ∈ 𝓩 𝒔.𝒕.𝒙𝟐 = 𝟖

Which of the following is a negation of ∀𝑥(𝑃(𝑥) → 𝑄(𝑥))? a. ∃𝒙(𝑷(𝒙)&~𝑸(𝒙)) b. ∀𝑥(𝑃(𝑥) → ~𝑄(𝑥)) c. ∀𝑥(~𝑃(𝑥)𝑜𝑟 𝑄(𝑥)) d. ∃𝑥(~𝑃(𝑥) → ~𝑄(𝑥))

a. ∃𝒙(𝑷(𝒙)&~𝑸(𝒙))

What is the logical relationship between the following two statements: ∃𝑦 ∈ 𝐵 𝑠.𝑡.∀𝑥 ∈ 𝐴,𝑃(𝑥,𝑦) ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑠.𝑡.𝑃(𝑥,𝑦) a. ∃𝒚 ∈ 𝑩 𝒔.𝒕.∀𝒙 ∈ 𝑨,𝑷(𝒙,𝒚) logically implies ∀𝒙 ∈ 𝑨 ∃𝒚 ∈ 𝑩 𝒔.𝒕.𝑷(𝒙,𝒚) but not the converse other way around. b. ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑠.𝑡.𝑃(𝑥,𝑦) logically implies ∃𝑦 ∈ 𝐵 𝑠.𝑡.∀𝑥 ∈ 𝐴,𝑃(𝑥,𝑦) but not the converse other way around. c. The two statements are logically equivalent. d. Neither statement logically implies the other.

a. ∃𝒚 ∈ 𝑩 𝒔.𝒕.∀𝒙 ∈ 𝑨,𝑷(𝒙,𝒚) logically implies ∀𝒙 ∈ 𝑨 ∃𝒚 ∈ 𝑩 𝒔.𝒕.𝑷(𝒙,𝒚) but not the converse other way around.

Which of the following inferences is invalid? a. 𝑷;𝑷 ∨ 𝑸/ ~𝑸 b. 𝐴&𝐵 / A c. ~𝑃;𝑃 ∨ 𝑄 / 𝑄 d. 𝐴;𝐴 → 𝐵 / B

a. 𝑷;𝑷 ∨ 𝑸/ ~𝑸

Which of the following is true: a. All valid arguments are sound. b. All sound arguments are valid. c. All valid arguments are factually correct. d. All factually correct arguments are valid.

b. All sound arguments are valid.

Every baseball stadium that Matt has been to is also one that Lauren has been to. Matt and Lauren decided that they wanted to write this fact down using symbolic logic. Matt wrote the following: Let 𝑀 be the set of baseball stadiums that Matt has attended, and 𝐿 be the set of baseball stadiums that Lauren has attended. Thus, 𝑀 ⊆ 𝐿. Lauren wrote the following: Let B be the set of baseball stadiums, 𝐴(𝑥,𝑦) = "𝑥 𝑎𝑡𝑡𝑒𝑛𝑑𝑒𝑑 𝑦", m= Matt, l= Lauren. Thus, ∀𝑦 ∈ 𝐵,𝐴(𝑚,𝑦) → 𝐴(𝑙,𝑦). Who is correct? a. Neither of them are correct. b. Both of them are correct. c. Matt is correct, but Lauren is not. d. Lauren is correct, but Matt is not.

b. Both of them are correct.

1. Ash, Misty, Brock and Tracy enter the Safari Zone and spot four different pokemon. Each trainer catches one of them. The pokemon are: spearrow, growlithe, oddish and poliwag. The latter three have a rock-paper-scissors relationship in that growlithe is super effective against oddish, oddish is super effective against poliwag, and poliwag is super effective against growlithe. Spearrow is not super effective against any of the others nor are any of the others super effective against spearrow. Ash caught either spearrow or growlithe, while Misty caught either oddish or poliwag. The pokemon Misty caught is super effective against the one Brock caught. If Brock caught poliwag, then Ash caught oddish. Which pokemon did Tracy catch? a. Growlithe b. Oddish c. Spearrow d. Poliwag

b. Oddish

1. There is an island of knights and knaves. Knights only say things that are true, while knaves only say things that are false. You are approached by two people. The first says: "Our identities are opposite one another." The second responds "our identities are the same." What can you conclude? a. The first one if a knave, the second one is a knight b. The first one is a knight, the second one is a knave c. Both are knights d. Both are knaves

b. The first one is a knight, the second one is a knave

The first US presidential election was held in 1792 and has been held every four years since, the most recent of which was in 2016. Which of the following sets is not the set of every year that has held a US presidential election? a. {1792 + 4𝑛}𝑛=0 𝑛=56 b. {𝒙|𝒙 + 𝟒𝒏,𝒙 = 𝟏𝟕𝟗𝟐,𝒏 ∈ 𝓝𝟎,𝒏 ≤ 𝟓𝟔} c. {1788 + 4𝑛|𝑛 ∈ 𝒩,𝑛 ≤ 57} d. {1792 + 4𝑛|𝑛 ∈ 𝒵,0 ≤ 𝑛 ≤ 56

b. {𝒙|𝒙 + 𝟒𝒏,𝒙 = 𝟏𝟕𝟗𝟐,𝒏 ∈ 𝓝𝟎,𝒏 ≤ 𝟓𝟔}

Which of the following is equal to {2,4,6,8,10,12}? a. {2 + 2𝑛}𝑛=1 𝑛=6 b. {𝟐𝒏}𝒏=𝟏 𝒏=𝟔 c. {𝑥 = 2𝑛|2 ≤ 𝑥 ≤ 12,𝑛 ∈ ℛ} d. {𝑥 + 2|2 ≤ 𝑥 ≤ 12}

b. {𝟐𝒏}𝒏=𝟏 𝒏=𝟔

Let G(x)= "x is a gerbil" F(x)= "x is fierce." Symbolize: Not all gerbils are fierce. a. ∀𝑥(𝐺(𝑥) → ~𝐹(𝑥)) b. ~[∀𝒙(𝑮(𝒙) → 𝑭(𝒙))] c. ~[∃𝑥(𝐺(𝑥)&𝐹(𝑥))] d. ∃𝑥(~𝐺(𝑥)&𝐹(𝑥)

b. ~[∀𝒙(𝑮(𝒙) → 𝑭(𝒙))]

Let C= "You will charge your car tonight," S= "You will get to school tomorrow," G= "You will stop at the gas station." Symbolize: "Unless you charge your car tonight, it will be necessary to stop at the gas station in order to get to school tomorrow" a. ~𝐶 → (𝐺 → 𝑆) b. ~𝑪 → (𝑺 → 𝑮) c. 𝐶 → (𝑆&~𝐺) d. 𝐶 → (𝐺 ∨ ~𝑆)

b. ~𝑪 → (𝑺 → 𝑮)

Let 𝐹 = {𝑇𝑒𝑑,𝐼𝑟𝑒𝑛𝑒,𝐸𝑢𝑛𝑖𝑐𝑒,𝐸𝑢𝑔𝑒𝑛𝑒}, B(x)= "x likes basketball," A(x)= "x likes the Avengers." Use the chart below to determine which of the following is false. Likes Basketball Likes the Avengers Ted Yes Yes Irene No Yes Eunice No No Eugene Yes Yes a. ∃𝑥 ∈ 𝐹(~(𝐵(𝑥) ∨ 𝐴(𝑥)) b. ∀𝒙 ∈ 𝑭(𝑩(𝒙) ∨ 𝑨(𝒙)) c. ∀𝑥 ∈ 𝐹(𝐵(𝑥) → 𝐴(𝑥)) d. ∃𝑥 ∈ 𝐹(𝐴(𝑥) → 𝐵(𝑥))

b. ∀𝒙 ∈ 𝑭(𝑩(𝒙) ∨ 𝑨(𝒙))

Let the universe for the following statement be all people. Let L = x loves y. How would you symbolize: "Everybody loves somebody." a. ∃𝑥∀𝑦,𝐿(𝑥,𝑦) b. ∀𝒙∃𝒚,𝑳(𝒙,𝒚) c. ∃𝑦∀𝑥,𝐿(𝑥,𝑦) d. ∀𝑦∃𝑥,𝐿(𝑥,𝑦)

b. ∀𝒙∃𝒚,𝑳(𝒙,𝒚)

What is in the place of the '$$' : 1) Suppose G 2) Suppose 𝐺 → 𝑆 3) Suppose (𝑆 ∨ 𝑅) → 𝑇 4) Show: T 5) By lines (1), (2) and modus ponens, we conclude $$ 6)It follows that, we have 𝑆 ∨ 𝑅 7) Thus, we can conclude T, because ... a. 𝑅 b. 𝑺 c. 𝑆 → 𝑅 d. 𝑆 ∨ 𝑇

b. 𝑺

Which of the following could fill in for ** given the following lines in a proof: 3) show: (𝐶⋁𝐷)&~𝐺 4) assume seeking a contradiction that ** a. 𝐶⋁𝐷 b. (~𝐶⋁~𝐷)&𝐺 c. (~𝑪 & ~𝑫)⋁𝑮 d. It could be any of these depending on what the premises were.

c. (~𝑪 & ~𝑫)⋁𝑮

Suppose 𝐴 ∩ 𝐵 = ∅,|𝐴| = 3,|𝐵| = 7. What is |𝐴 ∪ 𝐵|? a. 0 b. 4 c. 10 d. 21

c. 10

What is |𝐴 ∩ 𝐵|? a. {𝑓,𝑔,𝑤} b. {𝑓,𝑔} c. 3 d. 2

c. 3

How many subsets does {𝑎,𝑏,𝑐} have? a. 3 b. 7 c. 8 d. 9

c. 8

If George owns Boardwalk and George owns Park Place then George has a monopoly. George does not have a monopoly. What can you conclude: a. George owns either Boardwalk or Park Place but not both. b. George owns neither Boardwalk nor Park Place. c. George does not own Boardwalk or does not own Park Place. d. There is insufficient information to conclude any of the above.

c. George does not own Boardwalk or does not own Park Place.

If Katrina is fully immunized, then she has received a measles shot and she has received a mumps shot. Katrina is fully immunized. Katrina has received a measles shot. a. There is insufficient information to determine whether or not Katrina has received a mumps shot. b. Katrina has not received a mumps shot. c. Katrina has received a mumps shot. d. The premises are contradictory so any conclusion is meaningless.

c. Katrina has received a mumps shot.

Some humans are snakes. All snakes are reptiles. /Some reptiles are human. This is: a. Factually correct and valid. b. Factually correct but not valid. c. Valid, but not factually correct. d. Neither factually correct nor valid.

c. Valid, but not factually correct.

Which of the following is logically equivalent to ? a. ~A --> ~B b. A AND B c. ~A AND ~B d. ~A AND B

c. ~A AND ~B

Which of the following is logically equivalent to ~(A OR B) ? a. ~A OR ~B b. A AND B c. ~A AND ~B d. A <--> B

c. ~A AND ~B

Let P be the set of people, T be the set of things, 𝐶(𝑥) = "𝑥 𝑐𝑎𝑟𝑒𝑠," 𝐵(𝑦) = "𝑦 𝑤𝑖𝑙𝑙 𝑔𝑒𝑡 𝑏𝑒𝑡𝑡𝑒𝑟." Symbolize: "Unless somebody cares, nothing will get better." a. ∃𝑥 ∈ 𝑃 𝑠.𝑡.∀ 𝑦 ∈ 𝑇,𝐶(𝑥)&~𝐵(𝑦) b. ∀𝑥 ∈ 𝑃 ∀ 𝑦 ∈ 𝑇,~𝐶(𝑥) → ~𝐵(𝑦) c. ~[∃𝒙 ∈ 𝑷,𝑪(𝒙)] → [∀𝒚 ∈ 𝑻,~𝑩(𝒚)] d. [∃𝑥 ∈ 𝑃,𝐶(𝑥)] → [∀𝑦 ∈ 𝑇,𝐵(𝑦)]

c. ~[∃𝒙 ∈ 𝑷,𝑪(𝒙)] → [∀𝒚 ∈ 𝑻,~𝑩(𝒚)]

Let N(x) = "x is a natural born American" A(x)= "x is over 35 years old" P(x)= "x is eligible to become president." Symbolize: Only if one is a natural born American over the age of 35 years can one be eligible to be president. a. ∀𝑥((𝑁(𝑥)&𝐴(𝑥)) → 𝑃(𝑥)) b. ∃𝑥(𝑁(𝑥)&𝐴(𝑥)&𝑃(𝑥)) c. ∀𝒙(𝑷(𝒙) → (𝑵(𝒙)&𝑨(𝒙))) d. ∃𝑥((~𝑁(𝑥) ∨ ~𝐴(𝑥)) & 𝑃(𝑥))

c. ∀𝒙(𝑷(𝒙) → (𝑵(𝒙)&𝑨(𝒙)))

Let A be the set of actors {𝐴𝑚𝑦,𝐵𝑟𝑢𝑐𝑒,𝐶𝑙𝑎𝑟𝑒𝑛𝑐𝑒,𝐷𝑜𝑟𝑎} and 𝑀(𝑥,𝑦) = "x was in a movie with y." The following table shows you whether M is true (X) or false (blank) for given inputs x and y. Also, each person is symbolized by the first letter of their name. Which of the following statements is false? Amy Bruce Clarence Dora Amy X X X Bruce X X Clarence X X Dora X X X a. ∀𝑥 ∈ 𝐴,𝑀(𝑥,𝑥) b. ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴,(𝑀(𝑥,𝑦) → 𝑀(𝑦,𝑥)) c. ∀𝒚 ∈ 𝑨,(𝑴(𝒂,𝒚) → 𝑴(𝒃,𝒚)) d. ∃𝑦 ∈ 𝐴,(𝑀(𝑏,𝑦) & 𝑀(𝑑,𝑦))

c. ∀𝒚 ∈ 𝑨,(𝑴(𝒂,𝒚) → 𝑴(𝒃,𝒚))

In the following Venn Diagram, which of the following does the shaded region represent? a. (𝐴 ∩ 𝐵) ∪ 𝐶𝑐 b. 𝐶 − (𝐴 ∪ 𝐵) c. (𝐴𝑐 ∩ 𝐵𝑐) ∪ 𝐶 d. (𝑨 ∪ 𝑩) ∩ 𝑪𝒄

d. (𝑨 ∪ 𝑩) ∩ 𝑪𝒄

Let P= "I will write a proof" C= "I will write a counter-example" V= "The argument is valid." Symbolize: "If the argument is valid, I will write a proof; otherwise, I will write a counterexample." a. (𝑉 𝑜𝑟 ~𝑉) → (𝑃 𝑜𝑟 𝐶) b. 𝑉 ↔ 𝑃 & (𝑃 𝑥𝑜𝑟 𝐶) c. 𝑉 → 𝑃 → ~𝐶 d. (𝑽 → 𝑷)&(~𝑽 → 𝑪)

d. (𝑽 → 𝑷)&(~𝑽 → 𝑪)

Let D be the set of dogs, C be the set of collars and 𝑊(𝑥,𝑦) = "𝑥 𝑤𝑒𝑎𝑟𝑠 𝑦". What does ∃𝑦 ∈ 𝐶 𝑠.𝑡.∀𝑥 ∈ 𝐷,𝑊(𝑥,𝑦) symbolize? a. Some dog wears a collar. b. Some dog wears every collar. c. Every dog wears a collar. d. Every dog wears the same collar.

d. Every dog wears the same collar.

Some hippos are snakes. Some snakes are horses. /Some hippos are horses. This is: a. Factually correct and valid. b. Factually correct but not valid. c. Valid, but not factually correct. d. Neither factually correct nor valid

d. Neither factually correct nor valid

All firefighters are police officers. No firefighters are doctors. /No police officers are doctors. a. Factually correct and valid. b. Factually correct but not valid. c. Valid, but not factually correct. d. Neither factually correct nor valid.

d. Neither factually correct nor valid.

1. If Max is a Martian, then Max is a klug or Max is a charp. All Klugs are bex. Max is not a charp. You can conclude: a. Max is not a Martian, though there is insufficient information to conclude whether or not he is a klug or Bex. b. Max is a Martian, a klug and a Bex. c. Max is neither a Martian nor a klug, but could be a bex. d. There is insufficient information to determine whether or not Max is a Martian. However, if Max is a Martian, then Max is also a klug and a bex.

d. There is insufficient information to determine whether or not Max is a Martian. However, if Max is a Martian, then Max is also a klug and a bex.

1. Suppose that and . Furthermore, suppose that we know A is true. Of the three statements: A, B, C, how many are true? a. Insufficient information b. One c. Two d. Three

d. Three

Which of the following is not logically equivalent to the statement "Every natural number except for 1 is either prime or composite." a. ∀𝑥 ∈ 𝒩,𝑥 > 1 → (𝑥 𝑖𝑠 𝑝𝑟𝑖𝑚𝑒 ⋁𝑥 𝑖𝑠 𝑐𝑜𝑚𝑝𝑜𝑠𝑖𝑡𝑒) b. ∀𝑥 ∈ 𝒩,𝑥 ≠ 1 → (𝑥 𝑖𝑠 𝑝𝑟𝑖𝑚𝑒 ⋁𝑥 𝑖𝑠 𝑐𝑜𝑚𝑝𝑜𝑠𝑖𝑡𝑒) c. ∀𝑥 ∈ 𝒩 − {1},𝑥 𝑖𝑠 𝑝𝑟𝑖𝑚𝑒 ⋁ 𝑥 𝑖𝑠 𝑐𝑜𝑚𝑝𝑜𝑠𝑖𝑡𝑒 d. ∀𝒙 ∈ 𝓝,𝒙 ≠ 𝟏&(𝒙 𝒊𝒔 𝒑𝒓𝒊𝒎𝒆 ⋁𝒙 𝒊𝒔 𝒄𝒐𝒎𝒑𝒐𝒔𝒊𝒕𝒆)

d. ∀𝒙 ∈ 𝓝,𝒙 ≠ 𝟏&(𝒙 𝒊𝒔 𝒑𝒓𝒊𝒎𝒆 ⋁𝒙 𝒊𝒔 𝒄𝒐𝒎𝒑𝒐𝒔𝒊𝒕𝒆)

For the following chart, we will let 𝐹 be the set of friends, 𝑀 be the set of menu items. E(x,y) = "x is willing to eat y." Which of the following is false? Pizza Taco Salad Sandwich Burger Greg X X X X X Michelle X X Lisa X X Matt X X X X a. ∃𝑥 ∈ 𝐹 ∀𝑦 ∈ 𝑀, 𝐸(𝑥,𝑦) b. ∃𝑥 ∈ 𝐹 ∃𝑦 ∈ 𝐹, 𝐸(𝑥,𝑦) c. ∀𝑦 ∈ 𝑀 ∃𝑥 ∈ 𝐹, 𝐸(𝑥,𝑦) d. ∃𝒚 ∈ 𝑴 ∀𝒙 ∈ 𝑭, 𝑬(𝒙,𝒚)

d. ∃𝒚 ∈ 𝑴 ∀𝒙 ∈ 𝑭, 𝑬(𝒙,𝒚)

Let P be a propositional function on the set {1,2,3,4,5}. Suppose that you know that ∀𝑛 ∈ {1,2,3,4},𝑃(𝑛) → 𝑃(𝑛 +1). List all the possible truth sets for P.

{1,2,3,4,5},{2,3,4,5},{3,4,5},{4,5},{5},∅