List the members of these sets.
a) is a real number such that
b) is a positive integer less than 12
c) is the square of an integer and
d) is an integer such that
a)
b)
c)
d)
For each of these pairs of sets, determine whether the first
is a subset of the second, the second is a subset of the first,
or neither is a subset of the other.
a) the set of airline flights from New York to New Delhi,
the set of nonstop airline flights from New York to
New Delhi
b) the set of people who speak English, the set of people
who speak Chinese
c) the set of flying squirrels, the set of living creatures
that can fly
a) Every element in the second set is in the first set, so the second set is a subset of the first set. (the set of airline flights from new york to new delhi contains all nonstop flights + flights with layovers).
b) neither
c) the first is a subset of the second, and not vice versa.
Determine whether each of these pairs of sets are equal.
a)
b)
c) .
a) equal (duplicates and order dont matter)
b) not equal - first set has 1 element second has 2
c) not equal
Suppose that , , and are sets such that and . Show that .
Need to show that every element of is also an element of . Let . Since , we can conclude that . Since , the fact that implies , which is what we want to show.
Find the power set of each of these sets, where and
are distinct elements.
a)
b)
c)
a)
b)
c)
Prove that if and only if
We are proving an IFF so we have to show that and .
We must prove .
To prove that one set is a subset of another, we need to prove that an element being a member of one set implies membership of another set.
First Part: Prove
Suppose . , so , so , and hence .
Second Part: Prove
Suppose . Then for any we have , so , therefore . Thus
Let be a set. Show that .
By definition, consists of all pairs such that and . Since there are no elements , there are no such pairs, so . Similar reasoning shows that .
Subset take-away is a two player game involving a fixed finite set, . Players alternately choose proper, nonempty subsets of with the condition that one may not name a set
containing a set that was named earlier. A player who is unable to move loses.
For example, if is , then there are no legal moves and the second player wins. If is ,
then the only legal moves are and . Each is a good reply to the other, and so once again the
second player wins.
The first interesting case is when has three elements. This time, if the first player picks a subset
with one element, the second player picks the subset with the other two elements. If the first
player picks a subset with two elements, the second player picks the subset whose sole member
is the third element. Both cases produce positions equivalent to the starting position when has
two elements, and thus leads to a win for the second player.
Verify that when has four elements, the second player still has a winning strategy