a) The statement ∀n(n + 1 n) is true for all integers. For any integer n, adding 1 to it will always result in a number greater than n. Thus, for all integers in the domain, this statement holds true.

b) The statement ∃n(2n = 3n) is false. There does not exist an integer n such that n is equal to 3n. This can be verified by picking any integer and observing that the left-hand side is never equal to the right-hand side.

c) The statement ∃n(n = −n) is true. An example of such an integer n is 0, where n is to its additive inverse, −n.

d) The statementn(3n ≤ 4n) is true. For any integer n, multiplying 3 by n will always result in a number less than or equal to 4 multiplied by n. Therefore, this statement is true for all integers in the domain.

a) Everyone is studying discrete mathematics.

• True domain: The true domain could be a university where all the students are enrolled in discrete mathematics course.
• False domain: A false domain could be a general population where not everyone is studying discrete mathematics, such as a community with people of different professions and interests.

b) Everyone is older than 21 years.

• True domain: The true domain could be a bar or a location where only individuals over the age of 21 are permitted to enter or participate.
• False domain: The false domain could include a group of individuals that includes minors or younger individuals under the age of 21.

c) Every two people have the same mother.

• True domain: In a set of identical twins every two people have the same biological mother in the true domain.
• False domain: In a general group of people, not every two people have the same mother, thus constituting the false domain.

d) No two different people have the same grandmother.

• True domain: In a family unit where each individual has distinct grandmothers, the true domain would hold.
• False domain: In a scenario where cousins share the same grandmother from their mother's side, the false domain applies as two different people have the same grandmother.

a) Everyone speaks Hindi.

• True domain: A true domain could be a Hindi-speaking community where every member of the community speaks Hindi.
• False domain: A false domain could be a diverse international community where not everyone speaks Hindi.

b) There is someone older than 21 years.

• True domain: A true domain could be a workplace where all employees are above the age of years.
• False domain: A false domain could be a high school where all students are under the age of 21 years.

c) Every two people have the same first name.

• True domain: In a group identical twins, every two people have the same first name in the true domain. False domain: In a general population, not every two people have the same first name, thus constituting the false domain.

d) Someone knows more than two other people.

• True domain: In a large social network, it's likely that there is someone who knows more than two other people.
• False domain: In a small, closed community where individuals only interact with a few other people, it's possible that nobody knows more than two other people.