Assignment#1
Set Theory
1) Which of the following sets are equal?
2) List the elements of the following sets if the universal set is U = {a,
b, c, …, y, z}. Furthermore, identify which of the sets, if any, are equal.
3) Let
Which of the these sets can equal a set
under each of the following conditions?
(a) X and B are disjoint. (c) X ⊆ A but X
C.
(b) X ⊆ D but X
A.
B.
(d) X ⊆ C but X
4) Consider the universal set
Find:
and sets
(a)
and
(c)
(b)
and
(d)
and
(e)
and
,
,
and
(f)
and
5) The formula
defines the difference operation in terms of the operations of
intersection and complement. Find a formula that defines the union
in terms of the
operations of intersection and complement.
6) The Venn diagram shows sets
(a)
,
(b)
. Shade the following sets:
,
(c)
.
7) Write the dual of each equation:
(a)
(b)
8) Determine which of the following sets are finite:
(a) Lines parallel to the x axis.
(c) Integers which are multiples of 5.
(b) Letters in the English alphabet. (d) Animals living on the earth.
9) A survey on a sample of 25 new cars being sold at a local auto dealer was conducted to see
which of three popular options, air-conditioning (A), radio (R), and power windows (W), were
already installed. The survey found:
15 had air-conditioning (A),
5 had A and P ,
12 had radio (R),
9 had A and R,
3 had all three options. 11 had power windows (W),
4 had R and W ,
Find the number of cars that had: (a) only W ; (b) only A; (c) only R; (d) R and W but not A; (e)
A and R but not W ;
(f) only one of the options; (g) at least one option; (h) none of the options.
10) Find the power set
of
.
11) Given
(a) List the elements of
.
(b) Find
(c) Find the power set of
.
Relations
12) Let
13) Find
,
and
, and
. Find
where: (a)
(b)
14) Consider the relation
(a) Find the matrix
.
on
of .
(d) Draw the directed graph of .
(b) Find the domain and range of . (e) Find the composition relation ◦ .
(c) Find R−1.
(f) Find R◦R−1 and R−1◦R.
15) Let R be the relation on N defined by
, i.e.
(a) Write R as a set of ordered pairs. (c) Find R−1.
(b) Find the domain and range of R. (d) Find the composition relation R◦R.
16) Each of the following defines a relation on the positive integers N:
(1) “x is greater than y.”
(3)
(2) “xy is the square of an integer.” (4)
.
.
Determine which of the relations are: (a) reflexive; (b) symmetric; (c) antisymmetric; (d)
transitive.
.
Functions
17) Let
. Decide whether each set of ordered pairs is a function from
(a)
(b)
into
.
(c)
(d)
18) Let V = {1, 2, 3, 4}. For the following functions f : V → V and g: V → V , find:
(a) f ◦g; (b) g◦f ; (c) f ◦f :
f = {(1, 3), (2, 1), (3, 4), (4, 3)} and g = {(1, 2), (2, 3), (3, 1), (4, 1)}
19) Determine if each function is one-to-one.
(a) To each person on the earth assign the number which corresponds to his age.
(b) To each country in the world assign the latitude and longitude of its capital.
(c) To each book written by only one author assign the author.
(d) To each country in the world which has a prime minister assign its prime minister.
20) Let functions f, g, h from V = {1, 2, 3, 4} into V be defined by: f (n) = 6 − n, g(n) = 3,
h = {(1, 2), (2, 3), (3, 4), (4, 1)}. Decide which functions are:
(a) one-to-one; (b) onto;
(c) both;
(d) neither.
21) Decide which of the following functions are: (a) one-to-one; (b) onto; (c) both; (d) neither.
(1) f : Z2 → Z where f (n, m) = n − m;
(2) g: Z2 → Z2 where g(n, m) = (m, n);
(3) h: Z × (Z\0) → Q where h(n, m) = n/m;
(4) k: Z → Z2 where k(n) = (n, n).
22) Let f : R → R be defined by f (x) = 3x − 7. Find a formula for the inverse function f −1: R → R.
23) Consider permutations σ =(
Find: (a) τ ◦σ ; (b) σ ◦τ ; (c) σ 2; (d) σ −1; (e) τ −1
)
and τ = (
) in S6.