2.13: Exercises

Use an index card or a piece of paper folded lengthwise, and cover up the right-hand column of the exercises below. Read each exercise in the left-hand column, answer it in your mind, then slide the index card down to reveal the answer and see if you’re right! For every exercise you missed, figure out why you missed it before moving on.

Q	A
1. Is the set { Will, Smith } the same as the set { Smith, Will }?	Yes indeed.
2. Is the ordered pair (Will, Smith) the same as (Smith, Will)?	No. Order matters with ordered pairs (hence the name), and with any size tuple for that matter.
3. Is the set { { Luke, Leia }, Han } the same as the set { Luke, { Leia, Han } }?	No. For instance, the first set has Han as a member but the second set does not. (Instead, it has another set as a member, and that inner set happens to include Han.)
4. What’s the first element of the set { Cowboys, Redskins, Steelers }?	The question doesn’t make sense. There is no “first element" of a set. All three teams are equally members of the set, and could be listed in any order.
5. Let \(G\) be { Matthew, Mark, Luke, John }, \(J\) be { Luke, Obi-wan, Yoda }, \(S\) be the set of all Star Wars characters, and \(F\) be the four gospels from the New Testament. Now then. Is \(J \subseteq G\)?	No.
6. Is \(J \subseteq S\)?	Yes.
7. Is Yoda \(\in J\)?	Yes.
8. Is Yoda \(\subseteq J\)?	No. Yoda isn’t even a set, so it can’t be a subset of anything.
9. Is { Yoda } \(\subseteq J\)?	Yes. The (unnamed) set that contains only Yoda is in fact a subset of \(J\).
10. Is { Yoda } \(\in J\)?	No. Yoda is one of the elements of \(J\), but { Yoda } is not. In other words, \(J\) contains Yoda, but \(J\) does not contain a set which contains Yoda (nor does it contain any sets at all, in fact).
11. Is \(S \subseteq J\)?	No.
12. Is \(G \subseteq F\)?	Yes, since the two sets are equal.
13. Is \(G \subset F\)?	No, since the two sets are equal, so neither is a proper subset of the other.
14. Is \(\varnothing \subseteq S\)?	Yes, since the empty set is a subset of every set.
15. Is \(\varnothing \subseteq \varnothing\)?	Yes, since the empty set is a subset of every set.
16. Is \(F \subseteq \Omega\)?	Yes, since every set is a subset of \(\Omega\).
17. Is \(F \subset \Omega\)?	Yes, since every set is a subset of \(\Omega\), and \(F\) is certainly not equal to \(\Omega\).
18. Suppose \(X\) = { Q, \(\varnothing\), { Z } }. Is \(\varnothing \in X\)? Is \(\varnothing \subseteq X\)?	Yes and yes. The empty set is an element of \(X\) because it’s one of the elements, and it’s also a subset of \(X\) because it’s a subset of every set. Hmmm.
19. Let \(A\) be { Macbeth, Hamlet, Othello }, \(B\) be { Scrabble, Monopoly, Othello }, and \(T\) be { Hamlet, Village, Town }.What’s \(A \cup B\)?	{ Macbeth, Hamlet, Othello, Scrabble, Monopoly }. (The elements can be listed in any order.)
20. What’s \(A \cap B\)?	{ Othello }.
21. What’s \(A \cap \overline{B}\)?	{ Macbeth, Hamlet }.
22. What’s \(B \cap T\)?	\(\varnothing\).
23. What’s \(B \cap \overline{T}\)?	\(B\). (which is { Scrabble, Monopoly, Othello }.)
24. What’s \(A \cup (B \cap T)\)?	{ Hamlet, Othello, Macbeth }.
25. What’s \((A \cup B) \cap T\)?	{ Hamlet }. (Note: not the same answer as in item 24 now that the parens are placed differently.)
26. What’s \(A - B\)?	{ Macbeth, Hamlet }.
27. What’s \(T - B\)?	Simply \(T\), since the two sets have nothing in common.
28. What’s \(T \times A\)?	{ (Hamlet, Macbeth), (Hamlet, Hamlet), (Hamlet, Othello), (Village, Macbeth), (Village, Hamlet), (Village, Othello), (Town, Macbeth), (Town, Hamlet), (Town, Othello) }. The order of the ordered pairs within the set is not important; the order of the elements within each ordered pair is important.
29. What’s \((B \cap B) \times (A \cap T)\)?	{ (Scrabble, Hamlet), (Monopoly, Hamlet), (Othello, Hamlet) }.
30. What’s \(|A \cup B \cup T|\)?	7.
31. What’s \(|A \cap B \cap T|\)?	0.
32. What’s \(|(A \cup B \cup T) \times (B \cup B \cup B)|\)?	21. (The first parenthesized expression gives rise to a set with 7 elements, and the second to a set with three elements (\(B\) itself). Each element from the first set gets paired with an element from the second, so there are 21 such pairings.)
33. Is \(A\) an extensional set, or an intensional set?	The question doesn’t make sense. Sets aren’t “extensional" or “intensional"; rather, a given set can be described extensionally or intensionally. The description given in item 19 is an extensional one; an intensional description of the same set would be “The Shakespeare tragedies Stephen studied in high school."
34. Recall that \(G\) was defined as { Matthew, Mark, Luke, John }. Is this a partition of \(G\)? { Luke, Matthew } { John }	No, because the sets are not collectively exhaustive (Mark is missing).
35. Is this a partition of \(G\)? { Mark, Luke } { Matthew, Luke }	No, because the sets are neither collectively exhaustive (John is missing) nor mutually exclusive (Luke appears in two of them).
36. Is this a partition of \(G\)? { Matthew, Mark, Luke } { John }	Yes. (Trivia: this partitions the elements into the synoptic gospels and the non-synoptic gospels).
37. Is this a partition of \(G\)? { Matthew, Luke } { John, Mark }	Yes. (This partitions the elements into the gospels which feature a Christmas story and those that don’t).
38. Is this a partition of \(G\)? { Matthew, John } { Luke } { Mark } \(\varnothing\)	Yes. (This partitions the elements into the gospels that were written by Jews, those that were written by Greeks, those that were written by Romans, and those that were written by Americans).
39. What’s the power set of { Rihanna }?	{ { Rihanna }, \(\varnothing\) }.
40. Is { peanut, jelly } \(\in \mathbb{P}\)({ peanut, butter, jelly }?	Yes, since { peanut, jelly } is one of the eight subsets of { peanut, butter, jelly }. (Can you name the other seven?)
41. Is it true for every set \(S\) that \(S \in \mathbb{P}(S)\)?	Yep.