Question text is in black, answers in blue.
Yes, that is the standard notation for "not equal" -- I'm a little surprised you don't recall it from high school. Two sets are not equal if there is an element in one that is not in the other. More importantly, Problem 1.1.4 has no part (e) -- you appear to be looking at Exercise 1.1.4, which is not assigned for the homework because the answer is in the back of the book. Every regular section has both Exercises and Problems -- make sure that you are doing Problems for the homework.
Although your A is indeed a set, it is the same set as {1,3,4}, and the function f operates on sets written in the normal way with each element written only once. (Since f adds up one number for each element, it should not add up two different numbers for the same element as in your example.) So f(A) is really 2 + 8 + 16 = 26, and in fact there is no other set with that value of f.
First, you need to get the idea of a symmetric compound proposition. An example is the function from part (a), which is true when two of the four variables are true and false if zero, one, three, or four of them are true. If you know that a compound proposition is symmetric, and you know its value for some setting of the variables, then any other setting with the same number of true variables must have the same value. This means that you don't have to check all of the settings, just enough so that you know the values of the settings you didn't check.
No, you are making a type error. The way to compute f(∅) is to take all the numbers i such that i ∈ ∅, compute 2i for each, and add the results together. Since there are no such numbers, the sum equals 0. (Just as the answer to "what is the sum of the weights of all the unicorns in this room" is also 0.)
Write a boolean method. A "decision procedure"
for a language always inputs a string and then outputs a boolean,
which is true if the input string is in the language and
false if it is not.
The set statement is a valid set identity if it is true for all possible sets and all possible elements. This is true if and only if the corresponding compound proposition is a tautology (true for any truth values of the atomic propositions x ∈ A, x ∈ B, and X ∈ C). You can test whether you have a tautology by the method of truth tables from Section 1.6 -- doing this is Problem A-1. You may be able to argue more simply that you have a tautology. If you don't have a tautology, that is because at least one line of the truth table makes the compound proposition false. To answer 1.5.2, you need to find examples of sets A, B, and C and of an element x where this line of the truth table applies.
Last modified 30 Janary 2011