Question

Consider the interpretation with domain (1, 2, 3), where the extension of A(x) is 1, 2 and the extension of B(x) is 2, 3. Which of the following sentences are true in this interpretation?

a. Ǝxl-A(x) A-B(x))
b. Ǝx(-A(x) V-B(x))
c. Ǝx(A(x) A B(x))
d. Ǝx(A(x) - B(x))

Answer 1

In this given interpretation, sentence b and c are true, while sentence a and d are false.

Step-by-step explanation:

[a. Ǝx(A(x) ∧ ¬B(x))]

[b. Ǝx(¬A(x) ∨ B(x))]

[c. Ǝx(A(x) ∧ B(x))]

[d. Ǝx(A(x) ∧ ¬B(x))]

In the given interpretation, A(x) is true for elements 1 and 2, while B(x) is true for elements 2 and 3.

a. To determine if this sentence is true, we need to find an x value for which A(x) is true and B(x) is false. Since there is no such value in this interpretation, the sentence is false.

b. To determine if this sentence is true, we need to find an x value for which either ¬A(x) (not A(x)) or B(x) is true. Element 2 satisfies this condition since B(2) is true. Therefore, the sentence is true.

c. To determine if this sentence is true, we need to find an x value for which both A(x) and B(x) are true. Element 2 satisfies this condition since A(2) and B(2) are true. Therefore, the sentence is true.

d. To determine if this sentence is true, we need to find an x value for which A(x) is true and B(x) is false. Element 1 satisfies this condition since A(1) is true and B(1) is false. Therefore, the sentence is true.