请数学高手帮忙比较完整地给出下面一个题的答案

遥望王师又一年

Let A be a subset of the real numbers. Iva, Peter, Trefor, and Fedya tell us when
they like it:
? Iva likes A if and only if ?a ∈ R such that ?x ∈ A,a ≤ x.
? Peter likes A if and only if ?a ∈ A such that ?x ∈ A,a ≤ x.
? Trefor likes A if and only if ?a ∈ R such that ?x ∈ A,a < x.
? Fedya likes A if and only if ?a ∈ A such that ?x ∈ A,a < x.
Answer the following questions about Iva, Peter, Trefor, and Fedya. Prove your
answers.
(a) Is there any of them who likes every subset of the real numbers?
(b) Is there any of them who does not like any subset of the real numbers?
(c) Who likes the empty set?
(d) Are there two different people who like exactly the same subsets of the real
numbers?
(e) Is the following statement true?
For every A ? R, if Iva likes A then Peter likes A.
(f) Is the following statement true?
For every A ? R, if Peter likes A then Iva likes A.
Suggestion: Before you do anything else, try to understand with words which types
of subsets each of the four people like.

遥望王师又一年

刚才老师给出了答案。
Solution: As per the hint, we begin by describing the subsets of the real numbers that each person likes. This is not necessary, but it is going to make answering the following questions easier. We will say that a ∈ R is a lower bound for a subset A ? R if ?x ∈ A, a ≤ x (ie. a is less than or equal to each element of A). For example, 1 is a lower bound for [1,2] since it is true that 1 is less than or equal to each element of [1,2]. Similarly, 0 is also a lower bound for [1,2].
In addition, we will say that a ∈ R is a strict lower bound for A ? R if ?x ∈ A,a < x (ie. a is less than each element of A). Note that 1 is not a strict lower bound for [1,2], since 1 is not less than the element 1 of [1,2]. However, 0 is a strict lower bound for [1,2], as it is less than each element of [1,2]. Similarly, 1 is a strict lower bound of (1,2).
Using the terms defined above, let us describe the subsets A ? R that each person likes:
? Iva likes A if and only if it is possible to find a lower bound for A.
? Peter likes A if and only if it is possible to find a lower bound for A, such that the lower bound is itself an element of A.
? Trefor likes A if and only if it is possible to find a strict lower bound for A.
? Fedya likes A if and only if it is possible to find a strict lower bound for A, such that the strict lower bound is itself an element of A.
(a) Is there any of them who likes every subset of the real numbers?
Solution: No, none of them likes every subset of the real numbers. To prove this is to show that for each person, there is a subset of the real numbers that the person does not like.
I claim that none of them likes the subset (?∞,0). This is because it is impossible to find a lower bound for it, and also it is impossible to find a strict lower bound for it.
(b) Is there any of them who does not like any subset of the real numbers?
Solution: Yes, Fedya is the unique person that does not like any subset of the real numbers. To prove this, we must prove that Fedya does not like any subset of R, and for each of Iva, Peter, and Trefor, we must find a subset of the real numbers that the person likes.
Assume that Fedya likes a subset A ? R. By definition, this means that there exists some a ∈ A such that ?x ∈ A, a < x. Since a ∈ A, we can set x = a to obtain the statement a < a. This is a contradiction. So, Fedya cannot possibly like any subset A of R.
Now, consider the set [1,2]. Since 1 is a lower bound for [1,2], Iva likes it.
Noting that this lower bound is actually in [1,2], Peter also likes [1,2]. Finally,since 0 is a strict lower bound for [1,2], Trefor likes it.
(c) Who likes the empty set?
Solution: Iva and Trefor like the empty set, while Peter and Fedya do not like it. Let us prove this.
To show that Iva likes ?, we will show that 1 is a lower bound for ?. To check this, we must consider the statement ?x ∈ ?, 1 ≤ x (or equivalently, the
statement: if x ∈ ?, then 1 ≤ x). This statement is vacuously true, so that 1 is a lower bound for ?. Hence, Iva likes ?. By similar reasoning, 1 is also a strict lower bound for ?. We conclude that Trefor likes ?.
Now, look back at the conditions that determine whether Peter and Fedya like A ? R. For each condition to be satisfied, there must exist an element a ∈ A. Since ? does not have any elements, these conditions are never satisfied when A = ?. Hence, Peter and Fedya do not like ?.
(d) Are there two different people who like exactly the same subsets of the real numbers?
Solution: We claim that Iva and Trefor like exactly the same subsets of R, and that no other two people like exactly the same subsets. To prove our first
claim, we must show that if Iva likes A, then Trefor likes A, and then vice versa.
Assume that Iva likes A. By definition, there exists some a ∈ R such that ?x ∈ A, a ≤ x. Since a?1 < a, it is true that ?x ∈ A, a?1 < x. However, we
know that Trefor likes A if and only if there exists some number a 0 ∈ R (which need not be the same as the number a in Iva’s case!) such that ?x ∈ A, a 0 < x.
Hence, a 0 = a ? 1 satisfies Trefor’s condition for liking A, meaning that Trefor likes A.
Now, assume that Trefor likes A. By definition, there exists a ∈ R such that ?x ∈ A, a < x. Since a < x =? a ≤ x, we know that ?x ∈ A, a ≤ x. However,
we know that Iva likes A if and only if there exists a 0 ∈ R (which need not be the same as the number a in Trefor’s case!) such that ?x ∈ A, a 0 ≤ x. Hence, a 0 = a satisfies Iva’s condition for liking A, meaning that Iva likes A.
In conclusion, we have shown that Iva and Trefor like exactly the same subsets of R.
We now claim that Iva and Trefor are the only two people to like exactly the same subsets of R. To prove this, first recall that Fedya is the unique person
not to like any subset of the real numbers. In particular, no person and Fedya like exactly the same subsets of R. So, it just remains to prove that Peter does not like exactly the same subsets liked by Iva and Trefor. However, this follows from the fact that Peter does not like the empty set, while Iva and Trefor like it.
(e) Is the following statement true?
For every A ? R, if Iva likes A then Peter likes A.
Solution: This statement is not true. In other words, there exists a subset A ? R such that Iva likes A and Peter does not like A. To see this, just recall
that Iva likes ?, while Peter does not like ?.
(f) Is the following statement true?
For every A ? R, if Peter likes A then Iva likes A.
Solution: This statement is true. To prove it, we must show that if Peter likes A, then Iva likes A. So, assume that Peter likes A. By definition, there exists
some a ∈ A such that ?x ∈ A, a ≤ x. Since A is a subset of R and a ∈ A, we know that a ∈ R. The condition ?x ∈ A, a ≤ x then shows that A satisfies
Iva’s condition for liking A. In other words, Iva likes A, as desired.

遥望王师又一年

华人数学高手多的是，请感兴趣的同胞帮忙给出比较完整的答案，多谢。
正在多大上一年级的华人学生很多，许多学生最近做这个题，有些东西我还没彻底搞懂。

遥望王师又一年

我感到这次的题拿高分并不容易。
记得上个月开学时多大MAT137这门课的老师介绍这门课之后请约20位学过这门课的多大学生谈谈，大约20个学得好的学生，其中约15个是华人学生，我比较高兴。