Final answer:
In this question, we determine the truth value of various statements based on a domain of all integers. We analyze each statement and determine whether it is true or false by considering different scenarios and examples. The statements involve the existence of integers, inequalities, and equations.
Step-by-step explanation:
(a) The statement ∃n(2n=3n) is true if there exists an integer 'n' such that when '2n' is multiplied by '3', it is equal to '3n'. In this case, the statement is false because if we assume 'n' to be any integer, the equation doesn't hold true. For example, if we take 'n' as 1, then '2 x 1' is not equal to '3 x 1'.
(b) The statement ∃n(n=−n) is true if there exists an integer 'n' such that it is equal to its negative value. In this case, the statement is false because there is no integer 'n' that satisfies this condition. For any positive integer 'n', 'n' is not equal to its negative value.
(c) The statement ∀n(3n≤4n) is true if for any integer 'n', the inequality '3n≤4n' holds true. Since this inequality holds true for all integers, the statement is true.
(d) The statement ∀n(n²≥n) is true if for any integer 'n', the inequality 'n²≥n' holds true. This inequality holds true for all integers because when 'n' is a positive integer, 'n²' will always be greater than 'n'. So, the statement is true.
(e) The statement ∀n∃m(n²
(f) The statement ∃n∀m(n+m=0) is true if there exists an integer 'n' such that for all integers 'm', the sum 'n+m' is equal to 0. Since this condition is not possible as it would imply that 'n' can be any negative integer and 'm' can be any positive integer, the statement is false.
(g) The statement ∀n∃m(n+m=0) is true if for any integer 'n', there exists an integer 'm' such that their sum 'n+m' is equal to 0. This statement is true since for any integer 'n', we can choose 'm' as the negative of 'n' to satisfy the condition.
(h) The statement ∃n∃m(n² +m²=6) is true if there exist integers 'n' and 'm' such that their sum of squares is equal to 6. Since there are no integers that satisfy this condition, the statement is false.
(i) The statement ∃n∃m(n+m=4∧n−m=2) is true if there exist integers 'n' and 'm' such that their sum is equal to 4 and their difference is equal to 2. This statement is true, and we can find integers 'n' and 'm' that satisfy these conditions, for example, 'n' can be 3 and 'm' can be 1.
(j) The statement ∀n∀m∃p(p=(m+n)/2) is true if for all integers 'n' and 'm', there exists an integer 'p' such that it is equal to the average of 'n' and 'm'. This statement is true because for any 'n' and 'm', we can find the average of 'n' and 'm' which will be an integer.