Ma2201/CS2022
Quiz 0010
Discrete Mathematics
D Term, 2015
PRINT NAME:
SIGN :
1. (5 pts) Let X = {0, ∅}.
Circle each one of the following which is true.
a. 0 ⊆ X.
FALSE: 0 is not a set so it cannot be a subset.
b. ∅ ∈ P(P(X)).
TRUE: P(P(X)) is the set of all subsets of P(X). The empty set is a subset of every
set, so, regardless of what set P(X) is, ∅ ⊆ P(X), so ∅ ∈ P(P(X))
c. P(X) ⊆ P(P(X)).
FALSE: While is is true that P(X) ∈ P(P(X)), since every set is a subset of itself, in
order for P(X) ⊆ P(P(X)) to be true, every element of P(X) would have to be an element
of P(P(X)), which is to say, every subset of X would have to be a subset of P(X).
{0} ∈ P(X). But as in part a), 0 6⊆ X, so 0 6∈ P(X), so {0} 6∈ P(P(X)).
d. P(P(∅)) ⊆ P(X).
TRUE: P(∅) = {∅}, so P(∅) ⊆ X, so P(P(∅)) ⊆ P(X)
e. P(P(P(X))) has 20 elements.
FALSE: For any finite set, the cardinality of its power set is a power of 2.
If you are curious which power of 2,
|P(P(P(X)))| = 2|P(P(X))| = 2(2
(|P(X)|) )
= 2(2
(good thing it was true/false.)
1 of 2
(2|X| ) )
= 2(2
(22 ) )
4
= 2(2 ) = 216 = 65, 536
2. (5 pts) Let J = {a, b, c, d, e, f, g, h, i, j}, and use the usual ordering of the letters to
define the bit vectors, so the subset {a, e} corresponds to 0000010001.
a) What is the 100’th subset and what is its bit vector. (counting from 0.)
We convert 100 to binary. Using the iterated division method,
100/2
50/2
25/2
12/2
6/2
3/2
1/2
=
=
=
=
=
=
=
50 R 0
25 R 0
12 R 1
6R0
3R0
1R1
0R1
So 100 in binary is 1100100, and as a 10 digit number it is 0001100100 corresponding to
the set {c, f, g}
b) What is the 1000’th subset, what is its bit vector, and give the next 5 subsets
according to the lexicographic order on the bit vectors.
With the same method:
1000/2
500/2
250/2
125/2
62/2
31/2
15/2
7/2
3/2
1/2
=
=
=
=
=
=
=
=
=
=
500 R 0
250 R 0
125 R 0
62 R 1
31 R 0
15 R 1
7R1
3R1
1R1
0R1
Which is 1111101000 and already has ten digits, corresponding to {d, f, g, h, i, j}
The next five sets are:
1111101001 {a, d, f, g, h, i, j}
1111101010 {b, d, f, g, h, i, j}
1111101011 {a, b, d, f, g, h, i, j}
1111101100
{c, g, h, i, j}
1111101101 {a, c, d, f, g, h, i, j}
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