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Determine the truth-value of the following sentences. The truth-values of the simple sentences are given. Substitute the truth-values of the simple sentences into the Boolean sentence.

 

1. (20 Points/10 each) Determine the truth-value of the following sentences. The truth-values of the simple sentences are given. Substitute the truth-values of the simple sentences into the Boolean sentence.

• Michael Jordan played outfield for the White Sox. = TRUE
• Michael Jordan played guard for the Bulls. = TRUE
• Lebron James plays third base for the Indians. = FALSE
• Joan Benoit was an Olympic marathoner. = TRUE

a. ~[ (Michael Jordan played guard for the Bulls  ~Michael Jordan played guard for the Bulls) • Michael Jordan played outfield for the White Sox]  Lebron James plays third base for the Indians.

b. ~[~(Michael Jordan played outfield for the White Sox  Michael Jordan played guard for the Bulls) ≡ (Lebron James plays third base for the Indians • ~Joan Benoit was an Olympic marathoner)]

2. (20 points/10 each)
Translations of English sentences into Boolean truth-functional sentences:

i. Translate the sentence(s) into a truth-functional symbolic sentence.
ii. Produce a truth-table for the symbolic sentence.
iii. Please use the propositional variables assigned to the simple sentences (below).

R = Steve runs.
P = Ayca reads poetry.
B = Muk Yan plays badminton.

a. If Steve does not runs and Ayca reads poetry, then Muk Yan plays badminton and Steve runs.

b. Muk Yan plays badminton if and only if Ayca reads poetry, unless Steve does not run.

3. (70 points) Using a truth-table for the following symbolic argument:
i. Assess the validity of the argument.
ii. For each argument: conjoin the premises and imply the conclusion (i.e. create a conditional in which the antecedent is the conjunction of all the premises and the consequent is the conclusion). If your assessment is “valid” then this should result in a tautology, else not a tautology. (i.e. (P1 • P2 • .. • Pn)  ).

a. (35 points) 1. (p  q)  r
2. ~[(~p • ~q)  r]
 ~r

b. (35 points) 1. ~[~(p • q) • ~(~p  ~q)]
2. ~(~p  ~q)
 q
4. (40 points) Using a truth-table for the given symbolic sentences determine if the two sentences are logically equivalent. Remember to place the biconditional symbol between the two sentences, and derive the truth conditions for the biconditional.

a. (20 points)
i. [(p • q)  ~(p • q)]
ii. ~[ ~(p • q) • (p • q)]

b. (20 points)
i. p  q
ii. ~(p • q)  ~(p  q)
5. (50 points total)

• For the following completed valid proof:
o Determine the inference rule used for each line of the proof.
o Determine the line number(s) from the premises or the derived statements that justify the inference rule cited.
a) (30 points)

1. A  (B  D)
2. ~C  (D  E)
3. A  C
4. B
5. ~C
6.  E  D
7. ~A
8. B  D
9. D  E
10. B  E
11. (B  E) • (B  D)
12. B  B
13. E  D

b) (20 points)

1. C  D
2. ~D
3. ~C  ~E
4. E  F
 F
5. ~C
6. ~E
7. F

6. Bonus Section (20 points)

 For the following are valid arguments:
Derive a proof for each using only the basic nine.

a) (10 points)

1. A  B
2. ~A
3. B  (C  Z)
 C  (C • Z)
b) (10 points)

1. (B  D) • (A  (E  F))
2. B  A
3. ~D
4. ~F
 ~E  R

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