Question 1

Proposition 2I
Given the following proposition:
[∼(X ∨ B) ≡ (∼ Y ⊃ ∼ X)] ≡ ∼[(A ⊃ ∼ Y) • ∼(∼ X ⊃ ∼ B)]
-In Proposition 2I, the main operator is a:
A) Tilde.
B) Wedge.
C) Horseshoe.
D) Triple bar.
E) Dot.

Accepted Answer

The triple bar is the main operator in the proposition as it connects the two sides of the equivalence.

Question 2

Hertz runs a sale or both Avis buys new cars and Budget lowers rates.
A) H ∨ A • B
B) H ∨ (A • B)
C) (H ∨ A) • B
D) H • (A ∨ B)
E) H • A ∨ B

Accepted Answer

The statement specifies that Hertz runs a sale OR both Avis buys new cars and Budget lowers rates. Therefore, the correct logical operator to use is "OR" for the first part and "AND" for the second part. Option B correctly represents this as (H ∨ A • B).

Question 3

Not both Hertz runs a sale and Thrifty lays off workers if either Avis buys new cars or Budget lowers rates.
A) (∼H • ∼T) ⊃ (A ∨ B)
B) (A ∨ B) ⊃ (∼H • ∼T)
C) ∼(H • T) ⊃ (A ∨ B)
D) (A ∨ B) ⊃ ∼(H • T)
E) (A • B) ⊃ ∼(H ∨ T)

Accepted Answer

The correct logical representation of the statement "Not both Hertz runs a sale and Thrifty lays off workers if either Avis buys new cars or Budget lowers rates" is "If either Avis buys new cars or Budget lowers rates, then it is not the case that both Hertz runs a sale and Thrifty lays off workers." This is best represented by choice D, which states that if Avis buys new cars or Budget lowers rates (A ∨ B), then it is not the case that both Hertz runs a sale and Thrifty lays off workers (∼(H • T)).

Question 4

National does not improve efficiency or Enterprise does not reorganize, given that Thrifty lays off workers.
A) T ⊃ ∼(N ∨ E)
B) (∼N ∨ ∼E) ⊃ T
C) ∼(N ∨ E) ⊃ T
D) (T ⊃ ∼N) ∨ ∼E
E) T ⊃ (∼N ∨ ∼E)

Accepted Answer

The given statement can be translated to logical form as: If Thrifty lays off workers (T), then either National does not improve efficiency (∼N) or Enterprise does not reorganize (∼E). This is directly represented by choice E, "T ⊃ (∼N ∨ ∼E)."

Question 5

Statement 1I
Given the following statement:
(G • ∼ M) ⊃ (M ∨ G)
-Statement 1I is:
A) Consistent.
B) Logically equivalent.
C) Tautologous.
D) Contingent.
E) Self-contradictory.

Accepted Answer

To determine whether Statement 1I is tautologous, we can create a truth table for the statement.

G M ∼M G • ∼M M ∨ G (G • ∼M) ⊃ (M ∨ G) 
T T F F T T 
T F T T T T 
F T F F T T 
F F T F F T

In all possible combinations of truth values for G and M, the statement remains true. Therefore, it is a tautology.

Question 6

Statement 3I
Given the following statement:
[K • (P ∨ ∼ R)] • [K ⊃ (R • ∼ P)]
-The truth table for Statement 3I has how many lines?
A) Six.
B) Nine.
C) Eleven.
D) Eight.
E) Four.

Accepted Answer

To determine the number of lines needed in the truth table, we need to look at the number of distinct variables in the statement, which are K, P, and R. Each variable has two possible truth values (true or false), so the total number of possible combinations is 2^3 = 8. Therefore, the truth table for Statement 3I has eight lines.

Question 7

National's improving efficiency or Enterprise's reorganizing is a sufficient and necessary condition for Thrifty's laying off workers, provided that Avis buys new cars.
A) (N ∨ E) ≡ (A ⊃ T)
B) A ⊃ [(N ∨ E) ≡ T]
C) [(N ∨ E) ≡ T] ⊃ A
D) A ⊃ [(N ⊃ E) • (E ⊃ N)]
E) A ≡ [(N ∨ E) ⊃ T]

Accepted Answer

This is a conditional statement with multiple conditions. The best choice is (B) because it states that Avis buying new cars is necessary for the other conditions to lead to Thrifty's laying off workers. The statement also correctly represents that National's improving efficiency OR Enterprise's reorganizing is a sufficient and necessary condition for Thrifty's laying off workers. The other choices either don't represent the conditions correctly or don't take into account the necessity of Avis buying new cars.

Question 8

Statement 1I
Given the following statement:
(G • ∼ M) ⊃ (M ∨ G)
-The truth table for Statement 1I has how many lines?
A) Two.
B) Eight.
C) Four.
D) Six.
E) Nine.

Accepted Answer

Statement 1I involves two propositions, each of which could be either true or false. Thus, there are 4 possible combinations (TT, TF, FT, FF), and the truth table for Statement 1I should have 4 rows or lines.

Question 9

Statement 3I
Given the following statement:
[K • (P ∨ ∼ R)] • [K ⊃ (R • ∼ P)]
-Statement 3I is:
A) Valid.
B) Self-contradictory.
C) Tautologous.
D) Inconsistent.
E) Contingent.

Accepted Answer

The statement is self-contradictory because the two parts of the statement cannot both be true at the same time. The first part asserts K and either P or not R, while the second part asserts that if K then both R and not P, which contradicts the possibility of P or not R being true alongside K.

Question 10

Thrifty lays off workers only if neither National improves efficiency nor Enterprise reorganizes.
A) T ⊃ ∼(N ∨ E)
B) ∼(N ∨ E) ⊃ T
C) T ⊃ (∼N ∨ ∼T)
D) (∼N ∨ ∼E) ⊃ T
E) T ≡ ∼(N ∨ E)

Accepted Answer

The answer of Thrifty lays off workers only if neither...

Question 11

Hertz's running a sale and Budget's lowering rates are a necessary condition for either Avis's buying new cars or Thrifty's laying off workers.
A) (A • T) ⊃ (H ∨ B)
B) (H • B) ⊃ (A ∨ T)
C) (H • B) ∨ (A ∨ T)
D) (A ∨ T) ⊃ (H • B)
E) (H • B) ≡ (A ∨ T)

Accepted Answer

The statement "Hertz's running a sale and Budget's lowering rates are a necessary condition" can be represented as (H • B) ⊃, which means if H and B are true, then the following statement must also be true. Using the given options, only option D has the correct condition that Avis buying new cars or Thrifty laying off workers is a logical consequence of Hertz's sale and Budget's lower rates. Therefore, option D is the best choice.

Question 12

If National's improving efficiency implies that Enterprise reorganizes, then Avis buys new cars.
A) (N ⊃ E ⊃ A)
B) N ⊃ (E ⊃ A)
C) (N ⊃ E) ⊃ A
D) A ⊃ (N ⊃ E)
E) (E ⊃ N) ⊃ A

Accepted Answer

The statement can be translated as "If N, then if E then A". Therefore, the correct logic representation would be (N ⊃ E) ⊃ A.

Question 13

Budget lowers rates unless Hertz and Thrifty do not overhaul operations.
A) B ∨ (∼H • ∼T)
B) B ∨ ∼(H • T)
C) B ⊃ (∼H • ∼T)
D) (∼H • ∼T) ⊃ B
E) (B ∨ ∼H) • ∼T

Accepted Answer

The answer of Budget lowers rates unless Hertz and Thrifty...

Question 14

Proposition 1I
Given the following proposition:
[(B ≡ ∼ Y) • ∼(∼ X ⊃ A)] ∨ [∼(X ≡ A) ⊃ (Y • ∼ A)]
-In Proposition 1I, the main operator is a:
A) Dot.
B) Tilde.
C) Horseshoe.
D) Triple bar.
E) Wedge.

Accepted Answer

The answer of Proposition 1I
Given the following proposition:
[(B ≡ ∼...

Question 15

Thrifty's laying off workers is a sufficient condition for National's improving efficiency, if Enterprise reorganizes.
A) E ⊃ (N ⊃ T)
B) (T ⊃ N) ⊃ E
C) E ⊃ (T ⊃ N)
D) T ⊃ (N ⊃ E)
E) E ≡ (T ⊃ N)

Accepted Answer

The sentence is saying that if Thrifty's lays off workers, then National's improving efficiency is sufficient, but only if Enterprise reorganizes. In other words, E (Enterprise reorganizes) implies that if T (Thrifty's lays off workers), then N (National's improving efficiency). This is represented by option C.

Question 16

Proposition 2I
Given the following proposition:
[∼(X ∨ B) ≡ (∼ Y ⊃ ∼ X)] ≡ ∼[(A ⊃ ∼ Y) • ∼(∼ X ⊃ ∼ B)]
-Given that A and B are true and X and Y are false, determine the truth value of Proposition 2I.

Accepted Answer

Substituting the given values, we get:

[∼(false ∨ true) ≡ (∼ false ⊃ ∼ true)] ≡ ∼[(true ⊃ ∼ false) • ∼(∼ false ⊃ ∼ true)]

[¬true ≡ true] ≡ ¬[(true ⊃ true) • ¬(true ⊃ false)]

[false ≡ true] ≡ ¬[(true) • true]

false ≡ ¬true

false ≡ false

Therefore, Proposition 2I is true.

Question 17

Budget's lowering rates implies that Hertz runs a sale, only if Avis does not buy new cars.
A) (B ⊃ H) ⊃ ∼A
B) ∼A ⊃ (B ⊃ H)
C) (B ⊃ H) ≡ ∼A
D) (H ⊃ B) ⊃ ∼A
E) (B ⊃ H ⊃ A)

Accepted Answer

The answer of Budget's lowering rates implies that Hertz runs...

Question 18

Proposition 1I
Given the following proposition:
[(B ≡ ∼ Y) • ∼(∼ X ⊃ A)] ∨ [∼(X ≡ A) ⊃ (Y • ∼ A)]
-Given that A and B are true and X and Y are false, determine the truth value of Proposition 1I.

Accepted Answer

- (B ≡ ∼ Y) is true because A and B are true, and X and Y are false.
- ∼(∼ X ⊃ A) is false because X is false and A is true, making ∼ X true and A true, meaning (∼ X ⊃ A) is false, and ∼(∼ X ⊃ A) is true.
- Therefore, the first part of the proposition is true.
- (X ≡ A) is false because X is false and A is true.
- ∼(X ≡ A) is true.
- (Y • ∼ A) is false because Y is false and A is true.
- Therefore, the second part of the proposition is false.
- [∼(X ≡ A) ⊃ (Y • ∼ A)] is true because the antecedent is true and the consequent is false.
- The entire proposition is true because at least one of the two parts is true.

Question 19

Given the pair of statements: R ∨ S and S ⊃ ∼ R
These statements are:
A) Inconsistent.
B) Contradictory.
C) Invalid.
D) Logically equivalent.
E) Consistent.

Accepted Answer

R ∨ S means that at least one of R and S is true. 
S ⊃ ∼ R tells us that if S is true, then R is false. 
So, if S is true, then R must be false by the second statement. Therefore, the first statement must be true because at least one of R or S must be true. 
If S is false, then the second statement does not give us any information about the truth value of R, but the first statement is still true because R could be true or false and S is false. 
Therefore, these statements are consistent because it is possible for them to both be true at the same time.

Question 20

Statement 2I
Given the following statement:
(L • ∼ E) ≡ (E • ∼ L)
-Statement 2I is:
A) Self-contradictory.
B) Tautologous.
C) Consistent.
D) Contingent.
E) Inconsistent.

Accepted Answer

The answer of Statement 2I
Given the following statement:
(L • ∼...

Quiz 6: Propositional Logic

Proposition 2I Given the following proposition: [∼(X ∨ B) ≡ (∼ Y ⊃ ∼ X) ] ≡ ∼[(A ⊃ ∼ Y) • ∼(∼ X ⊃ ∼ B) ] -In Proposition 2I, the main operator is a:

Hertz runs a sale or both Avis buys new cars and Budget lowers rates.

Not both Hertz runs a sale and Thrifty lays off workers if either Avis buys new cars or Budget lowers rates.

National does not improve efficiency or Enterprise does not reorganize, given that Thrifty lays off workers.

Statement 1I Given the following statement: (G • ∼ M) ⊃ (M ∨ G) -Statement 1I is:

Statement 3I Given the following statement: [K • (P ∨ ∼ R) ] • [K ⊃ (R • ∼ P) ] -The truth table for Statement 3I has how many lines?

National's improving efficiency or Enterprise's reorganizing is a sufficient and necessary condition for Thrifty's laying off workers, provided that Avis buys new cars.

Statement 1I Given the following statement: (G • ∼ M) ⊃ (M ∨ G) -The truth table for Statement 1I has how many lines?

Statement 3I Given the following statement: [K • (P ∨ ∼ R) ] • [K ⊃ (R • ∼ P) ] -Statement 3I is:

Thrifty lays off workers only if neither National improves efficiency nor Enterprise reorganizes.

Hertz's running a sale and Budget's lowering rates are a necessary condition for either Avis's buying new cars or Thrifty's laying off workers.

If National's improving efficiency implies that Enterprise reorganizes, then Avis buys new cars.

Budget lowers rates unless Hertz and Thrifty do not overhaul operations.

Proposition 1I Given the following proposition: [(B ≡ ∼ Y) • ∼(∼ X ⊃ A) ] ∨ [∼(X ≡ A) ⊃ (Y • ∼ A) ] -In Proposition 1I, the main operator is a:

Thrifty's laying off workers is a sufficient condition for National's improving efficiency, if Enterprise reorganizes.

Proposition 2I Given the following proposition: [∼(X ∨ B) ≡ (∼ Y ⊃ ∼ X)] ≡ ∼[(A ⊃ ∼ Y) • ∼(∼ X ⊃ ∼ B)] -Given that A and B are true and X and Y are false, determine the truth value of Proposition 2I.

Budget's lowering rates implies that Hertz runs a sale, only if Avis does not buy new cars.

Proposition 1I Given the following proposition: [(B ≡ ∼ Y) • ∼(∼ X ⊃ A)] ∨ [∼(X ≡ A) ⊃ (Y • ∼ A)] -Given that A and B are true and X and Y are false, determine the truth value of Proposition 1I.

Given the pair of statements: R ∨ S and S ⊃ ∼ R These statements are:

Statement 2I Given the following statement: (L • ∼ E) ≡ (E • ∼ L) -Statement 2I is: