Question

If A ≣ (B v C) / ~C v B // A⊃B, what is the correct inference?

A) B ⊃ C
B) A ⊃ C
C) ~C
D) A ⊃ B

Answer 1

Step-by-step explanation:

To determine the correct inference from the given logical expression A ≣ (B v C) / ~C v B // A⊃B, we need to simplify and analyze the expression.

1. Let's simplify the expression step by step:

A ≣ (B v C) / ~C v B // A⊃B

A ≣ (B v C) / ~C v B // ~(A ^ ~B) [Using the material implication A⊃B ≡ ~(A ^ ~B)]

2. Next, let's simplify the expression inside the parentheses:

B v C [No further simplification possible]

3. Now, let's simplify the expression ~C v B using the rule of negation:

~C v B ≡ B v ~C [The OR operator is commutative]

4. Finally, let's simplify the entire expression:

A ≣ (B v C) / ~C v B // ~(A ^ ~B)

A ≣ (B v C) / B v ~C // ~(A ^ ~B)

From this simplification, we can see that the expression A is logically equivalent to B v C AND B v ~C AND ~(A ^ ~B).

5. Analyzing the simplified expression, we can infer the following:

- Since B v C is present in the expression A, we can infer that B ⊃ C is a valid inference.

- However, A ⊃ C is not a valid inference because C is not directly present in the simplified expression.

- ~C is not a valid inference as well because it is not directly present in the simplified expression.

- A ⊃ B is not a valid inference either since it is not directly present in the simplified expression.

Therefore, the correct inference from the given logical expression A ≣ (B v C) / ~C v B // A⊃B is:

A) B ⊃ C

Answer 2

The correct inference is A ⊃ B.

Step-by-step explanation:

The correct inference is D) A ⊃ B. To determine this, we can simplify the given expression step by step.

1. Distribute the negation (~) to the variables C and B: ~(B v C) / (~C v B)
2. Apply De Morgan's Law to the ~(B v C) term: (~B ^ ~C) / (~C v B)
3. Use the commutative law to rearrange the terms in the expression: (~B ^ ~C) / (B v ~C)
4. Combine the (~B ^ ~C) term: ~(B v C) / (B v ~C)
5. Apply the implication rule A ⊃ B: ~A v B / (B v ~C)
6. Use the commutative law to rearrange the terms: ~A v B / (~C v B)
7. Apply the implication rule A ⊃ B again: (~A v B) ⊃ (~C v B)

Therefore, the correct inference is A ⊃ B.