The statement "𝑥 + 𝑦 = 𝑥 − 𝑦" is only true when 𝑦 = 0. This is because the only way for 𝑦 to be added and subtracted from 𝑥 and still equal 𝑥 is if 𝑦 is zero.

a) 𝑄(1, 1) is false because 1 + 1 ≠ 1 - 1.

b) 𝑄(2, 0) is true because 2 + 0 = 2 - 0.

c) ∀𝑦𝑄(1, 𝑦) is false because the statement is not true for all values of 𝑦. It is only true when 𝑦 = 0.

d) ∃𝑥𝑄(𝑥, 2) is false because the statement is not true for any value of 𝑥 when 𝑦 = 2.

e) ∃𝑥∃𝑦𝑄(𝑥, 𝑦) is true because there exists at least one pair of values for 𝑥 and 𝑦 (specifically when 𝑦 = 0) that make the statement true.

f ) ∀𝑥∃𝑦𝑄(𝑥, 𝑦) is true because for every value of 𝑥, there exists a value for 𝑦 (specifically 𝑦 = 0) that makes the statement true.

g) ∃𝑦∀𝑥𝑄(𝑥, 𝑦) is true because there exists a value for 𝑦 (specifically 𝑦 = 0) that makes the statement true for all values of 𝑥.

h) ∀𝑦∃𝑥𝑄(𝑥, 𝑦) is false because the statement is not true for all values of 𝑦. It is only true when 𝑦 = 0.

i) ∀𝑥∀𝑦𝑄(𝑥, 𝑦) is false because the statement is not true for all values of 𝑥 and 𝑦. It is only true when 𝑦 = 0.

Question

The statement "𝑥 + 𝑦 = 𝑥 − 𝑦" is only true when 𝑦 = 0. This is because the only way for 𝑦 to be added and subtracted from 𝑥 and still equal 𝑥 is if 𝑦 is zero.

a) 𝑄(1, 1) is false because 1 + 1 ≠ 1 - 1.

b) 𝑄(2, 0) is true because 2 + 0 = 2 - 0.

c) ∀𝑦𝑄(1, 𝑦) is false because the statement is not true for all values of 𝑦. It is only true when 𝑦 = 0.

d) ∃𝑥𝑄(𝑥, 2) is false because the statement is not true for any value of 𝑥 when 𝑦 = 2.

e) ∃𝑥∃𝑦𝑄(𝑥, 𝑦) is true because there exists at least one pair of values for 𝑥 and 𝑦 (specifically when 𝑦 = 0) that make the statement true.

f ) ∀𝑥∃𝑦𝑄(𝑥, 𝑦) is true because for every value of 𝑥, there exists a value for 𝑦 (specifically 𝑦 = 0) that makes the statement true.

g) ∃𝑦∀𝑥𝑄(𝑥, 𝑦) is true because there exists a value for 𝑦 (specifically 𝑦 = 0) that makes the statement true for all values of 𝑥.

h) ∀𝑦∃𝑥𝑄(𝑥, 𝑦) is false because the statement is not true for all values of 𝑦. It is only true when 𝑦 = 0.

i) ∀𝑥∀𝑦𝑄(𝑥, 𝑦) is false because the statement is not true for all values of 𝑥 and 𝑦. It is only true when 𝑦 = 0.

Knowee AI · Accepted Answer