7. Classify each of the following as true or false. If false, tell why or give an example showing that it is not true.

(a) For all sets A and B, A – B = B – A

(b) For all sets A,   A

(c) For all sets A, A  A  

(d) The set {r, s, t,. . ., z} is a finite set.

(e) No set can be equivalent to all of its subsets

1. Try to post your question in ISO-8859-1 encoding so everyone sees what the special characters are.

2. You can type certain characters related to sets as follows:
"& c u p ;" ∪
"& c a p ;" ∩
"& s u b ;" ⊂
"& i s i n ;" ∈
"& n o t i n ;" ∉
"& e m p t y ;" ∅
"& a n d ;" ∧
"& o r ;" ∨
Do not include the double quotes nor the intervening spaces.

To determine whether each statement is true or false, we need to analyze and assess the validity of each statement individually. Here's a breakdown of each statement along with an explanation of each classification:

(a) For all sets A and B, A – B = B – A
This statement is generally false. The subtraction operation in set theory is not commutative. In other words, the order in which we subtract sets matters. To illustrate this, consider A = {1, 2} and B = {2, 3}. A - B = {1} while B - A = {3}. Hence, A - B is not equal to B - A.

(b) For all sets A, Ø ⊆ A
This statement is universally true. It states that the empty set (represented by Ø) is a subset of any set. Since the empty set does not contain any elements, it is trivially a subset of any set, including itself.

(c) For all sets A, A ⊆ A ∪ Ø
This statement is universally true. It states that any set A is always a subset of the union of A and the empty set. This is because the union of A and the empty set always includes all the elements of A, making A a subset of itself.

(d) The set {r, s, t, ..., z} is a finite set.
This statement is true. The set {r, s, t, ..., z} includes a finite number of elements since it contains a distinct list of letters from the alphabet ranging from r to z.

(e) No set can be equivalent to all of its subsets.
This statement is true. This is a result of Cantor's theorem, which states that for any set, the cardinality (size) of its powerset (set of all subsets) is strictly greater than the cardinality of the set itself. In other words, a set cannot have the same number of elements as its powerset, making it impossible for a set to be equivalent to all of its subsets.

By evaluating the properties and concepts discussed above, we can determine whether each statement is true or false.