Zimbabwe National Defence University
DMT 102 ENGINEERING MATHEMATICS 1 PAPER 2
Chapter 2
Author:
Dr J. Mushanyu
Department:
Aerospace Engineering
August 18, 2019
Chapter 1
Basic Concepts of Set Theory
1.1
The Importance of Set Theory
One striking feature of humans is their inherent need-and ability-to group objects according to
specific criteria. Our prehistoric ancestors grouped tools based on their hunting needs. They eventually evolved into strict hierarchical societies where a person belonged to one class and not another.
Many of us today like to sort our clothes at house, or group the songs on our computer into playlists.
The idea of sorting out certain objects into similar groupings, or sets, is the most fundamental
concept in modern mathematics. The theory of sets has, in fact, been the unifying framework for
all mathematics since the German mathematician George Cantor formulated it in the 1870s. No
field of mathematics could be described nowadays without referring to some kind of abstract set.
A geometer, for example, may study the set of parabolic curves in three dimensions or the set of
spheres in a variety of different spaces. An algebraist may work with a set of equations or a set
of matrices. A statistician typically works with large sets of raw data. And the list goes on. You
may have also read or heard that the most important unresolved problem in mathematics at the
moment deals with the set of prime numbers (this problem in number theory is known as Riemanns
Hypothesis; the Clay Institute will award a million dollars to whoever solves it.) As it turns out,
even numbers are described by mathematicians in terms of sets!
More broadly, the concept of set membership, which lies at the heart of set theory, explains how
statements with nouns and predicates are formulated in our language or any abstract language
like mathematics. Because of this, set theory is intimately connected to logic and serves as the
foundation for all of mathematics.
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1.2
What is a Set?
A set is a collection of objects called the elements or members of the set. These objects could be
anything conceivable, including numbers, letters, colors, even sets themselves! However, none of
the objects of the set can be the set itself. We discard this possibility to avoid running into Russells
Paradox, a famous problem in mathematical logic unearthed by the great British logician Bertrand
Russell in 1901.
1.3
Some Interesting Sets of Numbers
Let’s look at different types of numbers that we can have in our sets.
1. Natural Numbers
The set of natural numbers is {1, 2, 3, 4, . . . } and is denoted by N.
2. Integers
The set of integers is {. . . , −4, −3, −2, −1, 0, 1, 2, 3, 4, . . . } and is denoted by Z. The Z symbol
comes from the German word, Zahlen, which means number. Define the non-negative
integers {0, 1, 2, 3, 4, . . . } often denoted by Z+ . All natural numbers are integers.
3. Rational Numbers
The set of rational numbers is denoted by Q and consists of all fractional numbers i.e., x ∈ Q
if x can be written in the form pq , where p, q ∈ Z with q 6= 0.
4. Real Numbers
The real numbers are denoted by R.
5. Complex Numbers
The complex numbers are denoted by C.
1.4
Notation
1. A, B, C, . . . for sets.
2. a, b, c, . . . or x, y, z, . . . for members.
3. b ∈ A, if b belongs to A.
4. c ∈
/ A, if c does not belong to A.
5. ∅ is used for the empty set. There is exactly one set, the empty set or null set, which has
no members at all.
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6. A set with only one member is called a singleton or singleton set. for example, {x}.
1.5
Well-Defined Sets
A set is said to be well-defined if it is unambiguous which elements belong to the set. In other
words, if A is well-defined, then the question “Does x ∈ A?” can always be answered for any object
x.
For example, if we define C as the set of large numbers, then it is unclear which numbers should be
considered “large”. C is therefore not a well-defined set. Similarly, the set of all great Zimbabwean
footballers, or the set of all expensive restaurants in Harare, are also not well-defined.
1.6
Specification of Sets
1.6.1
Three Ways to Specify a Set
1. Listing all its members (List Notation).
2. By stating a property of its elements (Predicate Notation).
3. By defining a set of rules which generates (defines) its members (Recursive Rules).
List Notation
This is suitable for finite sets. It lists names of elements of a set, separated by commas and enclose
them in braces. For example
{2, 3, 5},
{a, b, d, m},
{Zimbabwe, SouthAfrica}.
Three Dot Abbreviation
For example, {1, 2, . . . , 100}.
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Predicate Notation
For example, {x|x is a natural number and x < 8}.
Reading : the set of all x such that x is a natural number and is less than 8.
The general form is {x|P (x)} where P is some predicate (condition, property).
Recursive Rules
For example, the set E of even numbers greater than 3,
(a) 4 ∈ E (b) if x ∈ E, then x + 2 ∈ E (c) nothing else belongs to E. The first rule is the
basis of the recursion, the second one generates new elements from the elements defined before and
the third rule restricts the defined set to the elements generated by (a) and (b).
The notion of a set does not allow for multiple instances (repetitions) of the same element in the
set, for example, {1, 2, 2, 3} is not a set.
The collection, out of which all sets under consideration may be formed, is called the universe of
discourse or universal set, denoted by U.
1.7
The Empty set (Null Set)
We have that the fundamental property of a set is that we can assert of each object whether or not
it is a member of the set.
Consider a set constructed by asserting of each object that it is not a member of the set. This set
has no members and is therefore called the empty set.
Definition 1.7.1. The null or empty set is the set that does not contain any elements, denoted
∅ = {} = {x|x 6= x}.
Example 1.7.1. (i) {x ∈ R|x2 = −1}
(ii) {x ∈ Z|x2 = 2}.
Theorem 1.7.1. There is exactly one empty set.
1.8
Identity and Cardinality
Two sets are identical if and only if (iff) they have the same elements or both are empty. So A = B
iff, for every x, x ∈ A ⇔ x ∈ B.
Example 1.8.1. {0, 2, 4} = {x|x is an even positive integer less than 5}.
The number of elements in a set A is called the cardinality of A, denoted by |A|. The cardinality
of a finite set is a natural number. Infinite sets also have cardinalities but they are not natural
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numbers. The set A is said to be countable or enumerable if there is a way to list the elements
of A.
1.9
Russell’s Paradox (antimony)
A paradox (antimony) is an apparently true statement that seems to lead to a logical selfcontradiction.
Its important to note that any given property, P (x) does not necessarily determine a set, i.e., we
cannot say that given any arbitrary property P , there corresponds a set whose elements satisfy the
property P .
Consider the following. There was once a barber man, wherever he lived, all of the men in this
town either shaved themselves or were shaved by the barber. And this barber man only shaved the
men who did not shave themselves. Did the barber shave himself?
Let’s say that he did shave himself. But from this he shaved only the men in town, who did not
shave themselves, therefore, he did not shave himself.
But we see that every men in town either shaved himself or was shaved by the barber. So he did
shave himself. We have a contradiction.
Russell observed that if S is a set, then either S ∈ S or S ∈
/ S, since a given object is either a
member of a given set or is not a member of that set. Consider the set of all sets that are not
members of themselves, R = {x|x is a set and x ∈
/ x}. R is an object, either R ∈ R or R ∈
/ R.
(i) Assume R ∈ R, then R is a set, and R ∈
/ R by definition.
(ii) Assume R ∈
/ R, then R ∈ R by definition of R, since we are assuming R is a set. But we
cannot have both R ∈ R and R ∈ R, so we reach a contradiction.
In both cases we have inferred the paradox that R ∈ R iff R ∈
/ R. In other words, the assumption
that R is a set has led to a contradiction and therefore there is no such thing, then, as the set of
all sets. To avoid unnecessary paradoxes, we assume the existence of the universal set, U. All this
leads to the following problems
1. There are things that are true in mathematics (based on assumptions).
2. There are things that are false.
3. There are things that are true that can never be proved.
4. There are things that are false that can never be disproved.
After this paradox was described, set theory had to be reformulated axiomatically as axiomatic
set theory.
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1.10
Inclusion
Definition 1.10.1. Having fixed our universal set, U, then for all x ∈ U. If A and B are sets (with
all members in U), we write A ⊆ B or B ⊇ A iff x ∈ A =⇒ x ∈ B. (⊆ , set inclusion symbol)
A set A is a subset of a set B iff every element of A is also an element of B. If A ⊆ B and A 6= B,
we call A a proper subset of B and write A ⊂ B.
Theorem 1.10.1. If A ⊆ B and B ⊆ C then A ⊆ C.
Proof. Let x ∈ A, then since A ⊆ B, we have x ∈ B and given that B ⊆ C, we conclude that
x ∈ C, thus A ⊆ C.
Example 1.10.1. (i) {a, b} ⊆ {d, a, b, e}
(iv) {a, b} 6⊂ {a, b}.
(ii) {a, b} ⊆ {a, b} (iii) {a, b} ⊂ {d, a, b, e}
Note that the empty set is a subset of every set, ∅ ⊆ A, for every set A and that for any set A, we
have A ⊆ A.
1.11
Axiom of Extensionality
Theorem 1.11.1. For any two sets A and B, A = B ⇐⇒ A ⊆ B and B ⊆ A.
1.11.1
Power Sets
The set of all subsets of A is called the power set of A and is denoted by P(A) and |P(A)| = 2|A|
where |A| is finite.
Example 1.11.1. If A = {a, b}, then P(A) = {∅, {a}, {b}, {a, b}}.
From the above example, a ∈ A, {a} ⊆ A, {a} ∈ P(A), ∅ ⊆ A, ∅ ∈
/ A, ∅ ⊆ P(A), ∅ ∈ P(A).
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1.12
Operations on Sets
1.12.1
Union and Intersection
Let A and B be arbitrary sets. The union of A and B, written A ∪ B, is the set whose elements
are just the elements of A or B or both.
A ∪ B := {x|x ∈ A or x ∈ B}.
Example 1.12.1. Let K = {a, b}, L = {c, d}, M = {b, d}, then K ∪ L = {a, b, c, d},
K ∪ M = {a, b, d}, L ∪ M = {b, c, d}, (K ∪ L) ∪ M = K ∪ (L ∪ M ) = {a, b, c, d}, K ∪ K = K,
K ∪ ∅ = ∅ ∪ K = K = {a, b}.
The intersection of A and B, written A ∩ B, is the set whose elements are just the elements of
both A and B.
A ∩ B := {x|x ∈ A and x ∈ B}.
Example 1.12.2. K ∩ L = ∅, K ∩ M = {b}, L ∩ M = {d}, (K ∩ L) ∩ M = K ∩ (L ∩ M ) = ∅,
K ∩ K = K, K ∩ ∅ = ∅ ∩ K = ∅.
1.13
Properties of ∪ and ∩
1. Every element x in A ∩ B belongs to both A and B, hence x belongs to A and x belongs to
B. Thus A ∩ B is a subset of A and of B i.e.,
A ∩ B ⊆ A and A ∩ B ⊆ B.
2. An element x belongs to the union A ∪ B if x belongs to A or x belongs to B, hence every
element in A belongs to A ∪ B and every element in B belong to A ∪ B, i.e.,
A⊆A∪B
and B ⊆ A ∪ B.
Theorem 1.13.1. For any sets A and B, we have (i) A ∩ B ⊆ A ⊆ A ∪ B and
(ii) A ∩ B ⊆ B ⊆ A ∪ B.
Theorem 1.13.2. The following are equivalent, A ⊆ B, A ∩ B = A, A ∪ B = B.
Proof. Suppose A ⊆ B and let x ∈ A. Then x ∈ B, hence x ∈ A ∩ B and A ⊆ A ∩ B. Then
A ∩ B ⊆ A. Therefore A ∩ B = A. Suppose A ∩ B = A and let x ∈ A. Then x ∈ (A ∩ B), hence
x ∈ A and x ∈ B. Therefore A ⊆ B.
Suppose again that A ⊆ B. Let x ∈ (A ∪ B), then x ∈ A or x ∈ B. If x ∈ A, then x ∈ B because
A ⊆ B. In either case x ∈ B. Therefore A ∪ B ⊆ B. But B ⊆ A ∪ B. Therefore A ∪ B = B. Now
suppose A ∪ B = B and let x ∈ A. Thus x ∈ (A ∪ B). Hence x ∈ B = A ∪ B, therefore A ⊆ B.
Definition 1.13.1. Two sets A and B are called disjoint sets if the intersection of A and B is
the null set i.e., A ∩ B = ∅.
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1.14
Difference and Complement
Definition 1.14.1. A minus B written A \ B or A − B, which subtracts from A all elements which
are in B (also called relative complement, or the complement of B relative to A) is defined as
A − B := {x|x ∈ A and x ∈
/ B}.
Example 1.14.1. K − L = {a, b}, K − K = ∅, K − M = {a}, K − ∅ = K, L − M = {c},
∅ − K = ∅.
1.14.1
Symmetric Difference
Definition 1.14.2. A 4 B = A ⊕ B := {x|x ∈ A or x ∈ B but not in both} or
A 4 B = A ⊕ B := (A ∪ B) \ (A ∩ B) = (A \ B) ∪ (B \ A).
The complement of a set A, is the set of elements which do not belong to A, i.e., the difference of
the universal set U and A. Denote the complement of A by A0 or Ac .
A0 = {x|x ∈ U and x ∈
/ A} or A0 = U − A.
Example 1.14.2. Let E = {2, 4, 6, . . . }, the set of all even numbers. Then E c = {1, 3, 5, . . . }, the
set of odd numbers.
1.15
Venn Diagrams
A simple and instructive way of illustrating the relationship between sets in the use of the so called
Venn-Euler diagrams or simply Venn diagrams.
1.16
Set Theoretic Equalities
1. Idempotent Laws (i) X ∪ X = X
(ii) X ∩ X = X.
2. Commutative Laws (i) X ∪ Y = Y ∪ X
(ii) X ∩ Y = Y ∩ X.
3. Associative Laws (i) (X ∪ Y ) ∪ Z = X ∪ (Y ∪ Z)
(ii) (X ∩ Y ) ∩ Z = X ∩ (Y ∩ Z).
4. Distributive Laws (i) X ∪(Y ∩Z) = (X ∪Y )∩(X ∪Z)
5. Identity Laws (i) X ∪ ∅ = X
(ii) X ∪ U = U
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(ii) X ∩(Y ∪Z) = (X ∩Y )∪(X ∩Z).
(iii) X ∩ ∅ = ∅
(iv) X ∩ U = X.
6. Complement Laws (i) X ∪ X c = U
(iv) X − Y = X ∩ Y c .
(ii) (X c )c = X
7. De Morgan’s Laws (i) (X ∪ Y )c = X c ∩ Y c
(iii) X ∩ X c = ∅
(ii) (X ∩ Y )c = X c ∪ Y c .
8. Consistency Principle (i) X ⊆ Y iff X ∪ Y = Y
(ii) X ⊆ Y iff X ∩ Y = X.
Example 1.16.1. Show that (Ac )c = A.
Proof. We need to show that A ⊆ (Ac )c and (Ac )c ⊆ A. Let x ∈ A then x ∈
/ Ac . If x ∈
/ Ac , then
x ∈ (Ac )c . By definition of subsets A ⊆ (Ac )c .
We want to show that (Ac )c ⊆ A. Let y ∈ (Ac )c , then y ∈
/ Ac . If y ∈
/ Ac , then y ∈ A. We have
c c
c c
shown that y ∈ (A ) =⇒ y ∈ A. Thus (A ) ⊆ A. By equality of sets (Ac )c = A.
Example 1.16.2. Show that A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C).
Proof. Let D = A ∩ (B ∪ C) and E = (A ∩ B) ∪ (A ∩ C). We have to prove first that D ⊆ E. Let
x ∈ D, then x ∈ A and x ∈ (B ∪ C). Since x ∈ (B ∪ C), either x ∈ B or x ∈ C or both. In case
x ∈ B we have x ∈ A and x ∈ B, so x ∈ (A ∩ B). On the other hand, if x ∈
/ B, then we must have
x ∈ C, so x ∈ (A ∩ C). Taking these two cases together, x ∈ (A ∩ B) or x ∈ (A ∩ C), so x ∈ E.
Now, we prove that E ⊆ D. Let x ∈ E. Suppose first that x ∈ (A ∩ B), then x ∈ A and x ∈ B, so
x ∈ A and x ∈ (B ∪ C), so x ∈ D. On the other hand, if x 6∈ (A ∩ B), then x ∈ (A ∩ C), so again
we obtain x ∈ A and x ∈ (B ∪ C), giving x ∈ D. Hence E ⊆ D. Hence both D ⊆ E and E ⊆ D
and we conclude that D = E and consequently A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C) .
1.17
Counting Elements in Sets
If A and B are disjoint sets, then
|A ∪ B| = |A| + |B|,
otherwise
|A ∪ B| = |A| + |B| − |A ∩ B|.
Example 1.17.1. Let A = {a, b, c, d, e} and B = {d, e, f, g, h, i}, so that A∪B = {a, b, c, d, e, f, g, h, i}
and A ∩ B = {d, e}. Since |A| = 5, |B| = 6, |A ∪ B| = 9, |A ∩ B| = 2, we have
|A ∪ B| = |A| + |B| − |A ∩ B| = 5 + 6 − 2 = 9.
1.18
The Algebra of Sets
We have considered the problem of showing that two sets are the same, however this technique
becomes tedious should the expressions involved be at all complicated. We shall develop an algebra
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of sets, to assist us in simplifying a given expression. The following basic laws are easily established.
Law 1 : (Ac )c = A Law 2 : A ∪ B = B ∪ A Law 3 : A ∩ B = B ∩ A
Law 4 : A ∪ (B ∩ C) = (A ∪ B) ∪ C Law 5 : A ∩ (B ∩ C) = (A ∩ B) ∩ C
Law 6 : A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C) Law 7 : A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)
Law 8 : (A ∪ B)c = Ac ∩ B c Law 9 : (A ∩ B)c = Ac ∪ B c Law 10 : U c = ∅
Law 11 : ∅c = U Law 12 : A ∪ ∅ = A Law 13 : A ∪ U = U Law 14 : A ∩ U = A
Law 15 : A ∩ ∅ = ∅ Law 16 : A ∪ Ac = U Law 17 : A ∩ Ac = ∅.
Example 1.18.1. By using the algebra of sets, show that A ∪ (B ∩ Ac ) = A ∪ B.
Proof.
A ∪ (B ∩ Ac ) = (A ∪ B) ∩ (A ∪ Ac ) by Law 6
= (A ∪ B) ∩ U by Law 16
= A ∪ B by Law 14.
1.19
Set Products
1.19.1
Ordered Pairs
Definition 1.19.1. Let n be any natural number and let a1 , a2 , . . . , an be any objects. Then
(a1 , a2 , . . . , an ) denotes the ordered n-tuple with first term a1 , second term a2 , . . . and nth term
an .
Example 1.19.1. (5, 7) denotes the ordered pair whose first term is 5 and second term 7. Note
that (5, 7, 2) is called an ordered triple, (5, 7, 2, 4) is called an ordered 4-tuple.
The fundamental statement we can make about an ordered n-tuple is that a given object is the kth
term of an ordered n-tuple.
Definition 1.19.2. Let A and B be any non-empty sets, then
A × B := {(a, b)|a ∈ A and b ∈ B}.
If A and B are both finite sets, then |A × B| = |A| · |B|. If A = B, we sometimes write A2 for
A × A.
Example 1.19.2.
1. If A = {1, 2} and B = {2, 3, 4}, then A×B = {(1, 2), (1, 3), (1, 4), (2, 2), (2, 3), (2, 4)}
and B × A = {(2, 1), (2, 2), (3, 1), (3, 2), (4, 1), (4, 2)}.
Notice that A × B 6= B × A, in general.
2. The Cartesian product R × R = R2 is the set of all ordered pairs of real numbers and this
represents the 2-dimensional Cartesian plane.
3. (s1 , t1 ) = (s2 , t2 ) if and only if s1 = s2 and t1 = t2 .
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1.20
Theorems on Set Products
Let A, B, C, D be sets, then
1. A × (B ∪ C) = (A × B) ∪ (A × C).
2. A × (B ∩ C) = (A × B) ∩ (A × C).
3. (A × B) ∩ (C × D) = (A ∩ C) × (B ∩ D).
4. (A × B) ∪ (C × D) ⊆ (A ∪ C) × (B ∪ D).
5. (A − B) × C = (A × C) − (B × C).
6. If A and B are non-empty sets, then A × B = B × A if and only if A = B.
7. If A1 ∈ P(A) and B1 ∈ P(B), then A1 × B1 ∈ P(A × B).
Example 1.20.1. Prove that (A ∪ B) × C = (A × C) ∪ (B × C).
Proof. Consider any element (u, v) ∈ (A ∪ B) × C. By definition u ∈ (A ∪ B) and v ∈ C. Thus
u ∈ A or u ∈ B. If u ∈ A, then (u, v) ∈ (A × C) and if u ∈ B, then (u, v) ∈ (B × C). Thus
(u, v) is in A × C or in B × C and therefore (u, v) ∈ (A × C) ∪ (B × C). This proves that
(A ∪ B) × C ⊆ (A × C) ∪ (B × C).
Now consider any element (u, v) ∈ (A × C) ∪ (B × C). This implies that (u, v) ∈ (A × C) or
(u, v) ∈ (B × C). In the first case u ∈ A and v ∈ C and in the second case u ∈ B and v ∈ C. Thus
u ∈ (A∪B) and v ∈ C which implies (u, v) ∈ (A∪B)×C. Therefore (A×C)∪(B×C) ⊆ (A∪B)×C.
Hence (A ∪ B) × C = (A × C) ∪ (B × C).
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