University Set Theory Homework: Problems, Solutions, and Analysis

This document provides a comprehensive set of solutions to various set theory problems. The assignment covers fundamental concepts such as set definitions, roster method, and identifying elements within sets. It explores set operations, including union, intersection, and Cartesian products, along with the properties of subsets, power sets, and the relationships between different sets. The solutions include detailed explanations, true/false evaluations, and descriptions of sets based on given conditions. The problems range in complexity, providing a solid understanding of set theory principles. Desklib is a platform where students can find this and other helpful study resources.

Set theory
Table of Contents
Introduction.............................................................................................................................................2
Question 2...............................................................................................................................................2
Question 6...............................................................................................................................................2
Question 8...............................................................................................................................................2
Question 14.........................................................................................................................................2
Question 22.........................................................................................................................................3
Question 28.............................................................................................................................................3
Question 37.............................................................................................................................................3
Question 42.............................................................................................................................................3
Question 50.............................................................................................................................................4
Question 58.............................................................................................................................................4
Question 63.............................................................................................................................................4
Question 68.............................................................................................................................................5
1

Set theory
Introduction
A set is a collection of distinct objects.
Question 2
a) True
b) False
c) True
d) True
Question 6
Description by roster method (listing the elements)
a) Natural numbers, the set is { 2,3}
b) Rational number, the set is { ∫7}
c) Natural numbers , the set is {4,-2)
Question 8
a) {2,3,4,6} includes the natural numbers.
b) {0,1,3,4}
c) X:X includes natural numbers that are in the form X> 4Y.
Question 14
R = {1, 3, π, 4.1, 9, 10}
T = {1, 3, π}
S = {{1}, 3, 9, 10}
U = {{1, 3, π}, 1}
a) The statement is true since 1 is included in S. For instance, 1 is a member of the S set.
b) Empty set is a subset of all set. Therefore the set S contains the subset of the empty set
however it does not contain any element of the subset since its already null.
c) The statement is true since the set T is included in the set U as a subset. The elements in
the set ({1, 3, π}) are included as members of a subset U.
d) The statement is true since the elements 1, 3, π are also included in the as members in the
both T and U sets.
e) The statement is false since the set T elements 1, 3, π are also listed as members of the set
R.
f) The statement is false since the set T elements 1, 3, π are also listed as members of the set
R, but not listed as subset of R since all the elements are listed openly.
g) The said statement is false since the set S (contains) has all the elements {1, 3, 9, 10}
but the set element 1 is included as a subset to the set S.
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Set theory
Question 22
a) The statement is false. The empty set is always included in all the sets as a subset
however the empty subset does not contain any element thus making the statement
false.
b) The statement is true since both sets contain zero elements, hence they are equal.
c) The statement is true since, if A is a proper/strict subset of B then means that the set
A has fewer elements than the set B, then set B is said to be a subset of the set C,
thus implying that the set A has fewer elements than the set C . The set A is a proper
subset of the set C.
d) The statement is true since if the set A doesn’t equal the set B and the set B does not
equal the set C, then it implies that the set A also does not equal the set C.
e) The argument is false since if the set A is contained as a member in the set B , and the
set B is a subset of the set C, it implies that the set B has fewer or equal elements to
the set C, therefore implying that the set A could possibly be a member of the set C.
Question 28
The powerset of the set S is the set of all subsets of set S.
P(S) = { ⱷ, {a}, {b}, {a,b}}
n[P(S)]= 4
Question 37
a) Y=3 ,X=1
b) X=8,Y=7
c) X=1, Y=4
Question 42
a) Unary operator since that a single operation is performed to produce a new value.
b) Binary operator since two operations are combined to produced single value.
Question 48
a) {r,v}
b) {w,u}
c) {p,q,r,s,t,v}
d) {q,r}
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Set theory
Question 50
a) {2,4}
b) {1,5,9}
c) {2,3,4,5,6,8}
d) {ⱷ}
e) {2,4,5}
f) {1,9}
g) {ⱷ}
Question 58
a) D’n B’
b) C n A’
c) D u C
d) Cn (AuB)’
Question 63
a) The statement is true since the union of the set A and set A is still the set A. for instance
the unions of set say A having elements {1,2,3,4} is still {1,2,3,4}.
b) The statement is true. Let y be an arbitrary element of BnB , then y is included in the set
B and also set b hence it implies that BnB is also B.
d)The argument is true since the double compliment of a set equal the set, for instance; let S
={1,2,3}, set A = {2} , then the compliment of the set A = {1,3}, further complimenting of
{1,3} = 2 which is the set A.
f. The argument is true since the intersection of A-B and B-A is empty set. A-B returns
values of the set A without any similar or common value of the set B, while the B-A
returns value of B without any similar or common value of the set A, this implies that
their intersection with result in an empty set since they’ve got common elements.
Question 68
Say that set A has elements { a,b,c} and set B has elements {1,2,3,4}
4

Set theory
a) the product of A and B will have a Cartesian form
Set a
Set b
1 2 3 4
A A,1 A,2 A,3 A,4
B B,1 B,2 B,3 B,4
C C,1 C,2 C,3 C,4
b)
A2
Set a
Set a
a b C
A A,a A,b A,c
B B,a B,b B,c
C C,a C,b C,c
c) B2
Set b
Set b
1 2 3 4
1 1,1 1,2 1,3 1,4
2 2,1 2,2 2,3 2,4
3 3,1 3,2 3,3 3,4
4 4,1 4,2 4,3 4,4
d) Max of AnB
The size of elements in a set depends on the order of the list
The maximum element in the set will therefore be the element ordered last in the list.
e) Min of AuB
The size of elements in a set depends on the order of the list
The minimum element in the set will therefore be the element ordered first in the list.
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Set theory
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