Elements of Mathematics
by
Francis Ssembatya
UMU
September 17, 2022
1
Sets
1.1
Definition
A set is a collection of objects or elements.
A set is always abbreviated with a capital letter. The elements or members of a
set are represented by lower case letters and are enclosed in braces { }.
A member of a set is denoted by the symbol ∈ . Hence x ∈ A means x is a member of
set A.
The negation of any symbol is obtained by drawing a slash (/) through the given
symbol. Thus x ∈
/ A reads x is not a member of set A.
1.2
Basic Concepts and Symbolism
There are two ways of representing sets
(a) Tabular notation / Listing members
(b) Set-builder notation.
1
1.2.1
The tabular notation
In the tabular notation method of set representation, all the members are listed and
enclosed in braces { }. Sometimes the tabular notation is known as the Roster method
of set representation.
Example 1.1. The set of the first nine natural numbers in tabular representation can
be written as {1, 2, 3, 4, 5, 6, 7, 8, 9}
Activity
Write the following in tabular notation.
A = {even numbers less than 10}
B = {x : x is a letter in the word MATHEMATICS}
C = {x : x is a digit in the number 7464235}
1.2.2
The set-builder notation
A set can also be represented by stating the properties or rules (formula) that the
members must satisfy.
Example 1.2. If the members of set N are all prime numbers then the set builder
notation is written as
N = {x : x is prime}.
Activity
1. Write the following in set-builder notation.
A = {a, e, i, o, u }
B = {3, 6, 9, 12, 15, ...}
D = {2, 3, 5, 7, 11, ...}
E = {1, 4, 9, 16, 25, ...}
2
2. List the members of the following sets
F = {x : x + 2 = 9}
B = {x : 1 ≤ x ≤ 7, x is an integer}
1.3
Types of sets
(a) Null set / Empty set:
A set is null or empty if it is without any element. We denote a null set by φ or
{}.
Example 1.3.
A = {x : 5x + 2 = 0, x is an integer} = φ
B = {x : x2 = −2, and x is a real number} = {}
(b) Non-empty set:
A set consisting of at least one element is said to be non-empty.
Example 1.4. A = {2, 3, 4} B = {2} C = {φ} are non-empty.
(c) Singleton set:
A set consisting of a single element is called a singleton set.
A = {2} B = {2} C = {φ} D = {0} E = {x : x is an integer satisfying the equation 2x+
5 = 9} are all singleton sets.
(d) Infinite sets:
A set is infinite if it contains an infinite number of elements. Elements of such
sets cannot all be listed.
Example 1.5. N = {1, 2, 3, 4, 5, ...} the set of natural numbers is infinite.
(e) Finite Sets: A set is finite if it has a finite number of elements. We have a
terminating count of the members in a finite set. A = {a, e, i, o, u} is a finite
set.
3
The cardinal number of a set is the number of distinct elements in that set. We
denote the cardinal number of set A by n(A) or |A|.
Example 1.6. If A = {1, 3, 4, 7} and B = {x, y, z} then n(A) = 4 and
|B| = 3.
(f) Subsets:
Set B is a subset of C if every element of B is in C. i.e for any x ∈ B we have
x ∈ C. This is usually denoted by B ⊆ C. Also C is called a superset of B and
we can also write C ⊇ B.
Example 1.7. If P = {2, 3} and Q = {1, 2, 3, 4} then P ⊆ Q.
Theorem 1.1. Given sets A, B, andC such that A ⊆ B and B ⊆ C then
A ⊆ C.
Proof:
Let x ∈ A. Since A ⊆ B then x ∈ B for all x ∈ A. But B ⊆ C ⇒ x ∈ C for all
x ∈ B. Therefore A ⊆ C.
Equality of Sets:
Two sets A and B are said to be equal denoted by A = B if and only if A ⊆ B
and B ⊆ A.
Definition 1.1. Let A and B be sets. If A ⊆ B and A 6= B, then A is said to
be a proper subset of B and we write A ⊂ B. Note that in this case there is at
least one element of B which is not in A.
Example 1.8. If P = {2, 3} and Q = {2, 3, 4} then P ⊂ Q.
If A = {4, 5} and B = {4, 6, 4} then A 6⊂ B.
Example 1.9. List down the possible subsets of A = {α, β, λ}.
The number of subsets of a set with n elements is 2n .
Every set is a subset of itself.
A null set is a subset of any set.
4
(g) The Power set:
The family of all subsets of any set B is called the power set of B and is denoted
by 2B .
Example 1.10. Given set A = {α, β, λ}, list down the members of 2A .
Example 1.11. Given set P = {α, β, λ, µ}, list down the power set of P.
(h) Disjoint sets:
Two sets A and B are said to be disjoint if they have no elements in common.
(i) Universal set:
A universal set ξ is a set of all elements under consideration. The universal set
is a superset of all sets under consideration.
(j) Set difference:
The difference of two sets A and B denoted by A − B is the set of elements of
Awhich are not in set B i.e A − B = {x : x ∈ A, x 6∈ B}.
Example 1.12. If A = {1, 2, 3, 4} and B = {1, 3, 6, 9} then A − B = {2, 4}.
(j) Set complement:
The set containing all elements of the universal set ε which are not in A is called
0
the complement of set A. This is denoted by A , Ac or A.
1.4
Venn diagrams and Operation on sets
All sets under consideration are put in a rectangular enclosure.
Example 1.13. Represent the following sets on a venn diagram. ε = {1, 2, 3, 4, 5, 6}
, A = {1, 2} , B = {1, 2, 3, 4}, C = {2, 4, 6}.
(a) Venn diagram for subsets
(b) Set union:
The union of two sets A and B is the set of all elements which are in A or B or
in both and is by A ∪ B i.e
A ∪ B = {x : x ∈ A or x ∈ B}
5
(c) Set Intersection:
The intersection of two sets A and B is the set of all elements which are both in
A and B we denote this by A ∩ B i.e
A ∩ B = {x : x ∈ A and x ∈ B}
Note that
1. Sets A and B are disjoint if A ∩ B = φ.
2. For any two intersecting sets
n(A ∪ B) = n(A) + n(B) − n(A ∩ B).
(d) Symmetric difference:
The symmetric difference of two sets A and B denoted by A 4 B is defined by
A 4 B = (A − B) ∪ (B − A)
Example 1.14. Given A={1, 2, 3, 4, 5} and B={0, 1 ,2, 6} find A 4 B.
(e) Set complements: Venn diagram for two and three joint sets.
Example 1.15.
0
0
1. Given the following; n(A) = 13, n(B) = 8, n(A ∩ B ) = 2
0
0
and n(ε) = 20. Find n(A ∩ B), n(A ∩ B), n(B ∩ A) and n(A ∪ B).
0
2. Given the following; n(A) = 18, n(B) = 11, n(A ∩ B) = 4 and n(ε) = 24.
0
0
0
Find n(A ∩ B), n(A ∩ B ), n(B ∩ A) and n(A ∪ B).
0
3. Given the following; n(A) = 20, n(B) = 10, n(B ∩ A) = 15 and n(ε) = 32.
0
0
0
Find n(A ∩ B), n(A ∩ B ), n(A ∩ B) and n(A ∪ B).
Example 1.16. In a class all 50 students take Maths, 18 take Chemistry, 17
take Biology and 24 take Physics. Of those taking three subjects, 5 take Physics
and chemistry, 7 take Physics and Biology and 6 take chemistry and Biology.
Only two of the students take all the four subjects. How many of them take only
maths?
6
1.4.1
Special sets
(a) Natural Numbers N:
The natural numbers are positive whole numbers i.e N ={1, 2, 3, 4, 5, ...}
(b) Integers Z:
The set of positive and negative whole numbers including a zero is called the set
of integers i.e Z ={...-4, -3, -2,-1, 0, 1, 2, 3, 4, 5, ...}
Notice that N ⊂ Z
(c) Rational numbers Q: Numbers that can be expressed as ratios of two integers
are called rational numbers.That is
p
Q = {x : x = , gcd(p, q) = 1, where p, q ∈ Z}
q
Every integer is a rational number thus N ⊆ Z ⊂ Q.
(d) Irrational numbers:
a
Numbers which cannot be expressed in the form where the gcd(a, b) = 1 are
b
√ √
called irrational numbers e.g 2, 3, π.
(e) Real Numbers R:
Real numbers refer to numbers that can be represented by points on a straight
line. A collection of all other numbers mentioned above is a set of real numbers.
They are denoted by R
N⊂Z⊂Q⊂R
(f) Complex Numbers C:
Any number in the form a + bi where a and b are real numbers and i =
√
−1
is known as a complex number. Since any complex number can be expressed in
the form a + 0i then the set of Complex numbers is a super set of other sets of
numbers.
N⊂Z⊂Q⊂R⊂C
7
Example 1.17.
Z1 = 2 + 3i
Z2 = −4 + 2i
W1 = 2 − 3i
W2 = −2 − 2i
1.4.2
Closure of a set under an operation
A set on numbers is said to be closed under a mathematical operation of
(a) Addition if the summation of any two elements in the set give an element in the
set.
(b) Subtraction if the difference between any two elements in the set give an element
in that set.
(c) Multiplication if the product of any two elements in the set is in that set.
(d) Division if for any two elements in the set the quotient is in that set.
1.5
Ordered pairs
For two numbers (elements) x and y we define the ordered pair (x, y) as a pair in
which x is the first element and y the second element.
1.5.1
Properties of ordered pairs
Let x, y, r, z be elements
(i) (x, y) 6= (y, x) when x 6= y.
(ii) For (x, y) = (r, z) then x = r and y = z.
Example:
Given that (y, y + 2x)= (2, 4) find the values of x and y.
8
1.6
Cartesian Product
Let A and B be two non-empty sets. We defined the Cartesian product of A and B
denoted by A × B by A × B = {(a, b)|a ∈ A and b ∈ B}.
Note that members in the cartesian product are ordered pairs.
For three non empty sets, we define the cartesian product as
A × B × C = {(a, b, c)|a ∈ A and b ∈ B, and c ∈ C}.
For A = |{2, 3} B = {0, 1, 4} find
(i) A × B
(ii) B × A
(iii) A × A
(iv) n(A × B)
1.6.1
Properties of Cartesian Product
(a) A × B 6= B × A
(b) If A = φ and B = φ then A × B = φ.
(c) If n(A) = p and n(B) = q then n(A × B) = pq
If A = {1, 2} B = {0, 1, 3} find
(i) (A ∪ B) × (B − A)
(ii) B × (B − A)
(iii) (A ∪ B) × (A 4 B)
9
1.7
Laws of Set theory
1. Distributive law
Let A, B and C be sets then
(a) A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)
(b) A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)
2. De-Morgan,s Law
0
0
0
0
0
0
(a) (A ∩ B) = (A ∪ B
(b) (A ∪ B) = (A ∩ B
2
Matrices
A matrix is a rectangular array of numbers in horizontal rows and vertical columns.
Each entry in the matrix is called an element.
Consider


6 −7 6 2










A = 5 14 5 −2








0 2 3 −3
is a matrix with 3 rows and 4 columns.
2.1
Order of Matrices
The number of rows and columns are used to describe the order of a matrix.
Examples:
10
Matrix
Order of the matrix

(i)

4 6 3


−2 0 1
2 by 3 or (2 × 3)
(ii)


0 6 4 −1




2 7 −2 0 


3 3 −1 6
3 by 4 or (3 × 4)


12




(iii)  20 


−40
(iv)
3 by 1 or (3 × 1)
h
i
120 −130 −160 173
(1 × 4)
Activity:
Write down examples of matrices with the following order:
(a) 1 × 1
(b) 5 × 3
(c) 3 × 5
(d) 4 × 4
2.2
Special Matrices
(i) Row Matrix
A matrix with only one row is called a row matrix.
Examples:
11
B
=
h
i
7 0 4 5
M
=
h
i
0 0 0
N
=
h
i
250 −300
(ii) Column matrix
A matrix with only one column is called a column matrix.
Examples:
 
4
 
6
P
=
R


1000




2000


= 

4000


6000


−25




Q = −30


0
(iii) Square matrix
A square matrix has the same number of rows and columns.
Examples:
12


4 −22

D = 
0 8

E

−2




= −5 0
0


3 −4 1
0
3

F
is a 2 × 2 square matrix
is a 3 × 3 square matrix

100
200 −350 500




−250 350
450
550 
 is a 4 × 4 square matrix
= 


 730 −740 810 −913


−610 913 −876 337
(iv) Null (Zero) matrix
A matrix which has every element zero (0) is called a zero or null matrix.
Examples:
S


0 0




= 0 0


0 0
(v) Identity (Unit) matrix
A unit matrix or an identity matrix is a square matrix with each element in
the leading diagonal 1 and every other element 0.
Examples:
13


1 0

D = 
0 1
E
is the 2 × 2 identity matrix


1 0 0




= 0 1 0


0 0 1

1 0 0


0 1 0
F = 

0 0 1

0 0 0
These matrices are
is the 3 × 3 identity matrix

0


0
 is the 4 × 4 identity matrix

0

1
denoted by I2 , I3 and I4 respectively.
Activity:
Write down the
(a) 5 × 5
(b) 6 × 6 identity matrices.
(vi) Diagonal matrix
A diagonal matrix 
is a
4


zero. The matrices 0

0
squarematrix whose non-main diagonal entries are all


0 0

−3 0

 are both diagonal matrices.
3 0  and 

0 0
0 −2
(vii) Triangular matrix
A triangular matrix is defined as a square matrix whose non-zero elements are
either above or below the diagonal.The matrices


−3 0 0




−2 3 0


7 4 1
14
and


0 0 8




0 1 8


0 0 5
are both triangular matrices.
(viii) Matrix Transpose
Given matrix A, the transpose of A denoted by AT is another matrix obtained
by interchanging rows with columns.
(ix) Symmetric and Anti-symmetric Matices
2.3
Addition and Subtraction of matrices
We can add or subtract matrices if they are of the same order. Addition and subtraction is done by considering corresponding elements.
Example: 

4 −3

Given A = 
4 −1


2 −3

B=
0 6

and
0
4


C=
−3 4
find:
(i) A + B
(ii) B − C
(iii) A + B + C
(iv) A + C − B
Solution:
(i) A + B =


4 −3


4 −1
Hence A + B =
(ii) B − C =
+


2 −3


0 6
=


4 + 2 −3 + −3


4 + 0 −1 + 6
=


6 −6

.
4 5


6 −6


4 5


2 −3


0 6

−
0
4



−3 4
15

=

−3 − 4


0 − −3 6 − 4
2−0
=


2 −7

.
3 2
Hence B − C
(iii) A + B + C
=


2 −7


3 2


4 −3

= 
4 −1


2 −3


0 6
+


4 + 2 −3 + −3

= 
4 + 0 −1 + 6


6 −6


4 5
=

Hence A + B + D
(iv) A + D − B
6

−2

0

−3
=


4 −3


4 −1
=
+
+

+

+


0
4



−3 4
0
4
−3 4
4

.

.

−3 4
1


9

4

4

−


2 −3


0 6

4+0−2
−3 + −3

= 
4 + −3 − 0 −1 + 4 − 6
Hence A + D − B
0

+

0 4

.
−3 4


2 4


1 −3
=
Activity:


20 25 −30




Given M = 10 20 −20


20 25 30
find:


−40




N =  20
30
50 


−30 −40 60
10

20
(i) M + N
(ii) N − P
16
and

−15




P = 20 −25 30 


35 40
45
5
10
(iii) P + M + N
(iv) (P − N ) − (N − M )
2.4
2.4.1
Multiplication of matrices
Multiplication of a matrix by scalar quantity
The product of a matrix by a scalar quantity is conducted by multiplying each element
by the scalar.
Example:


−4




Given P=  6 2 −4


10 20 0
0
2
find:
(i) 2P
(ii) −4P
(iii) − 12 P
Solution:
17

(i)
2P

0 2 −4




2 ×  6 2 −4


10 20 0
=


2 × 0 2 × 2 2 × −4




=  2 × 6 2 × −2 2 × 0 


2 × 10 2 × 20 2 × 0


−8




∴ 2P =
12 −4 0 


20 40 0


0 2 −4




(ii) −4P = −4 ×  6 2 −4


10 20 0
0
4

−4P
2.4.2
=

0
−8 16




−24 8
0


−40 −80 0
Multiplication of two matrices
Multiplication of any two matrices A and B is compatible if the number of columns
of the matrix on the left is equal to the number of rows of the matrix on the right.
Examples:


1 2

(1) Let A = 
2 3


2 4

B=
−1 2
find:
(i) AB
(ii) BA
(iii) B 2
18


2 6

(2) Given P = 
4 2


4 4
1




(3) If A = 0 1
3


1 −2 −2
 
1
 
 
Q = 6
 
3

find PQ.

2 4




B = −1 2


−2 2
find AB.
Exercise:


1 2
4




If A = 2 3 −5


2 −4 4


2
4
0




B =  2 −2 −1


−2 3
1
find:
(i) AB
(ii) BA
(iii) Show that BI = IB = B
2.5
Determinants
Determinant of a 2 × 2 matrix


a b
 the determinant of A denoted by |A| or det A
Given a 2 × 2 square matrix A = 
c d
is defined as
2.5.1
|A| = ad − bc
or
det A = ad − bc.
Example:
19
Find the determinant of the following matrices


1 2

(i) A = 
2 3


10 −2

(ii) B = 
5 −3


3 −3

(iii) F = 
0 −2

(iv) P = 
2.5.2
10 220
21
33


Determinant of a 3 × 3 matrix
Given a 3 × 3 matrix


a b c




B = d e f 


g h j
the determinant of B is defined as
detB = a
e f
−b
h j
d f
g j
Example:
Compute the determinant of the following matrices
20
+c
d e
g h


2 1 0




(i) A = 3 2 1


1 1 2


2 −3 5




(ii) B = 1 0 3


2 1 3
2.5.3
Properties of Determinants
The following properties may be used in finding the determinant of a matrix.
1. |A| = |AT |.
2. If two rows or columns of a matrix are equal then the determinant of the matrix
is zero i.e |A| = 0.
3. If any row or column in a matrix is zero then |A| = 0.
4. |AB| = |A||B|.
5. The determinant of a triangular or diagonal matrix is given by the product of
the diagonal elements. That is, if


2 3 7




A = 0 6 4


0 0 1
then |A| = 2 × 6 × 1 = 12.
Exercise:
Using properties of determinants find the determinants of the following matrices;


2 1

(i) A = 
0 0
21


2 1 0




(ii) B = 0 2 3 


0 0 −2


1
0
0 0




−1 2
0 0


(iii) C = 

−3 −4 −5 0


4
3
1 2
2.6
Matrix Inverse
The inverse of square matrix A is another matrix denoted by A−1 such that
AA−1 = A−1 A = I
where I is an identity matrix.


a b
 then A−1 =
If A = 
c d

1
detA
d
× (AdjA) where Adjunct or Adjoint A = 
−c
Example:
Find the inverse of


2 1
 and verify that BB −1 = B −1 B = I.
1. B = 
3 2


5 −1
 and verify that C −1 C = I.
2. C = 
3 7


−2 6
 and verify that T T −1 = I.
3. T = 
−3 4
22
−b
a

.
2.7
Simultaneous equations using matrix methods
Examples:
Solve the following simultaneous equations by the matrix method
(1)
2x + 3y = 8
x − 2y = −3.
(2)
3x + 4y = 18
7x + y = 7.
(3)
3y − 4x = 9
y + x = 3.
(4) Paul bought 5 sackets of washing powder and an apple at Shs 1,700 from ABC
Super-market. Alice bought 15 sackets of washing powder and 2 apples at Shs
4,400 from the same super-market. Find the price of each item.
23
2.8
Application
(1) The table below shows the number of books and pens used by two students John
and Mary during Semester I and Semester II 2017.
Student
John
Mary
Semester I
Semester II
Books Pens Books Pens
5
4
4
3
3
5
7
2
(a) Write down a 2 × 2 matrix to represent the number of items bought in
(i) Semester I
(ii) Semester II
(b) Use matrix addition to find the number of items bought by the two students
in the academic year 2017.
(c) If the cost of a book is Shs.4000/ = and that of a pen is Shs.700/ =, write
down a 2 × 1 matrix for the prices of the commodities.
(d) Use matrix multiplication to find the total expenditure by each of the two
students on the items bought.
(2) Alex bought 2 books and one pen while Grace bought one book and 3 pens from
supermarket A. From supermarket B Alex bought 4 books and a pen while Grace
bought 6 books and 2 pens. A pen costs Shs. 500 and a book costs Shs. 1000.
(i) Represent the items bought by each student from the two supermarkets
using 2 × 2 matrices.
(ii) Use matrix multiplication to find the total expenditure of each student on
the items bought.
3
Functions
A function f from set A to another set B denoted by f : A → B assigns to each
member of A a unique number B.
1. If x ∈ A and f : A → B then we write y = f (x) where y ∈ B.
24
2. y is called the image of x under function f.
3. Each member of A has one and only one unique image in B.
4. The set A is called the domain of the function f.
5. Set B is called the range (co-domain) of f.
6. Elements in the domain set A are usually called objects
Sketch graphs of f (x) = x2 and f (x) = x3 .
3.1
Functional values
1. If f (x) = 3x2 − 2x + 5 find the value of f (0) f (−1) f (a) f (x − β)
2. If f (x) =
x2
1
find the value of f (0) f (2) f (−1) f ( a1 )
−1
3. If f (x) = 2x2 + ax + 2 and that f (1) = 8 determine the value of a
3.2
Composite functions
Given f (x) = 2 − x and g(x) = x2 + 1 find f (0) f g(x) f g(−1) gf (x) gf (−2)
3.3
Equality of functions
Given two functions f : A → B and g : A → B, we say that the function f is equal
to function g if
f (x) = g(x) ∀x ∈ A
3.4
One-to-one functions
We say that a function f : A → B is one-to-one if the functional values
f (x1 ) = f (x2 )
imply that
x1 = x2
∀x1 , x2 ∈ A
25
Alternatively, we say that a function f is one-to-one if
x1 6= x2
implies that
f (x1 ) 6= f (x2 )
3.4.1
Horizontal line test
A function is one-to-one if and only if no horizontal line intersects its graph more than
once.
1. Is the function g(x) = x2 one-to-one?
2. Is the function g(x) = x3 one-to-one?
3.5
Inverse functions
The function f −1 is called the inverse of function f.
Examples:
• Given that f (x) = 2x − 3 find
(i) f −1 (x)
(ii) f −1 (2)
• Find the inverse of g(x) =
• Given that h(x) =
4
2x − 4
2x − 3
find h−1 (x).
x+1
3.6
Types of functions
3.6.1
Polynomial functions
A polynomial of degree n is defined as
P (x) = an xn + an−1 xn−1 + an−2 xn−2 + ... + a1 x + a0
The expression x2 + 4x + 5 is a polynomial of degree 2.
26
for n 6= 0
3.6.2
Trigonometric functions
Trigonometric functions are functions that involve trigonometric ratios such as
• f (x) = cosx
• g(x) = sinx
3.6.3
Exponential functions
These are functions taking the form f (x) = ax . A special popular case of exponential
function is f (x) = ex where e = 2.7183
3.6.4
Logarithmic functions
These are of the form f (x) = logx.
Example
Given that f (x) = 4ex + 5logx find f (2).
4
4.1
Mathematical Logic
Propositions/Statements
A proposition/Statement is a sentence that is either true or false but not both at the
same time.
Examples of expressions which are propositions include;
Masaka is a new city in Uganda.
30 + 47 = 120.
Kampala is 190 Km away from Mbarara.
Makerere is is a public university in Uganda.
Examples of expressions which are not propositions include;
How old is Tom?
27
3x ∈ Z.
He is honest.
4.1.1
Types of Propositions/Statements
There are two types of propositions namely ; Simple and Compound Propositions.
4.1.2
Simple Propositions
A simple proposition is the one without any connective like “ and”, “or”, “ if ....then”,
etc. Examples of simple propositions include;
The number 3 is a factor of 15.
Boston is in USA
The sum of two odd numbers is even.
The number 5 is prime.
4.1.3
Compound Propositions
A compound proposition is formed when two or more simple propositions are joined
by logical connectives to form one proposition.
Example 4.1. Peter failed the test and Grace is happy.
Table 1: Logical connectives
Connective Symbol
Name
not
∼
Negation
and
∧
Conjunction
or
∨
Disjunction
If ....then
→
Conditional/ Implication
if and only if
↔
Bi-conditional
Examples of how connectives work;
1. Let
p : John is sick,
28
q : Jane is happy.
Then we have
∼p:
p∧q :
p∨q :
p→q :
p↔q :
2. Symbolically write the following propositions
I like IT and hate Business studies.
If I get 2000 dollars, I will attend a conference in UK.
Sets A and B are equal if and only if A ⊂ B and B ⊂ A.
4.2
Truth Assignment and Truth Tables
The truth or falsity of a given proposition is called its truth value. We denote a true
proposition by T and a false proposition by F.
4.2.1
Truth Tables
A truth table presents the truth values of one or more propositions for all possible
combinations of the truth values of each possible combinations.
4.2.2
Truth Table for Negation
If p is any proposition, then its negation is written ∼ p which reads “ not p”.
If the truth value of p is T , then the truth value of ∼ p is F. Thus
Table 2:
p ∼p
T F
F T
29
4.2.3
Truth Table for Conjunction
The compound proposition p ∧ q has truth value T whenever both p and q have the
same truth value T otherwise it has truth value F.
Table 3: Truth
p
T
T
F
F
Table for Conjunction
q p∧q
F
F
T
T
Note that the “ and” connective is commutative.
4.2.4
Truth Table for Disjunction
The proposition p ∨ q has truth value F only when p and q have the same truth value
F otherwise it is true.
Table 4: Truth
p
T
T
F
F
Table for Disjunction
q p∨q
F
F
T
T
Note that the truth value of p ∨ q is the same as that of q ∨ p.
4.2.5
Truth Table for Conditional/Implication
The proposition p → q will have truth value F only when a true statement implies a
false proposition i.e when p is true and q is false. Otherwise p → q has truth value T.
30
Table 5: Truth Table
p
T
T
F
F
4.2.6
for the Conditional connective
q p→q
F
F
T
T
Truth Table for Bi-conditional
The bi-conditional p ↔ q is true when p and q are both true or both false.
Table 6: Truth Table for the Conditional connective
p
T
T
F
F
4.3
q p→q
F
F
T
T
q→p
p↔q
Compound Propositions with a number of logical connectives
Example 4.2. If I plan well and the weather is not bad then I will attend the party.
(a) Write the statement in symbolic form.
(b) Write the truth table for the given compound statement.
Example 4.3. Write the truth table for
(i) (p ∨ q) ∧ s
(ii) s → (p ∨ q)
4.4
Tautology
We define a tautology as a compound proposition which is always true regardless of
the truth values of its simple propositions.
Example 4.4. Check whether the following compound propositions are tautologies.
31
(i) q∨ ∼ p
(ii) ∼ p ∨ (p ∨ q)
(iii) (p ∧ q) → (p ∨ q)
(iv) p ∧ (p ∧ q)
4.5
Contradiction
A compound proposition which is always false whatever be the truth values of its
components is called a contradiction.
Example 4.5. Show that the following statements are contradictions
(i) Jane is fine and Jane is not fine.
(ii) (p ∧ ∼ q) ∧ (p ∧ ∼ q)
Example 4.6. Prove the following DeMorgan’s laws
(i) (A ∪ B)0 = A0 ∩ B 0
(ii) (A ∩ B)0 = A0 ∪ B 0
32