CS137-8

DISCRETE
MATHEMATICS
MODULE 3:
SETS
 CHAPTER SUMMARY
• Sets
• The Language of Sets
• Set Operations
• Set Identities
• Functions
• Types of Functions
• Operations on Functions
• Computability

• Sequences and Summations

• Types of Sequences
• Summation Formulae
SETS
Section 2.1
 SECTION SUMMARY
• Definition of sets
• Describing Sets
• Set-Builder Notation
• Some Important Sets in Mathematics
• Empty Set and Universal Set
• Subsets and Set Equality
• Cardinality of Sets
• Tuples
• Cartesian Product
 SETS
• A set is an unordered collection of objects.
• the students in this class
• the chairs in this room
• The objects in a set are called the elements, or members of the

• The notation a ∈ A denotes that a is an element of the set A.

set. A set is said to contain its elements.

• If a is not a member of A, write a ∉ A

 DESCRIBING A SET: ROSTER
METHOD
• S = {a,b,c,d}

S = {a,b,c,d} = {b,c,a,d}
• Order not important

• Each distinct object is either a member or not; listing more than

S = {a,b,c,d} = {a,b,c,b,c,d}
once does not change the set.

• Ellipses (three-dots) may be used to describe a set without listing all of

the members when the pattern is clear.
S = {a,b,c,d, … ,z }
 ROSTER METHOD

V = {a,e,i,o,u}
• Set of all vowels in the English alphabet:

• Set of all odd positive integers less than 10:

O = {1,3,5,7,9}
• Set of all positive integers less than 100:
S = {1,2,3,……..,99}
Set of all integers less than 0:
S = {…., -3,-2,-1}
•
 SOME IMPORTANT SETS
N = natural numbers = {0,1,2,3….}
Z = integers = {…,-3,-2,-1,0,1,2,3,…}
Z⁺ = positive integers = {1,2,3,…..}
R = set of real numbers
R+ = set of positive real numbers
C = set of complex numbers.
Q = set of rational numbers
 SET-BUILDER NOTATION

S = {x | x is a positive integer less than 100}

• Specify the property or properties that all members must satisfy:

O = {x | x is an odd positive integer less than 10}

O = {x ∈ Z⁺ | x is odd and x < 10}

S = {x | P(x)}
• A predicate may be used:

• Example: S = {x | Prime(x)}

Q+ = {x ∈ R | x = p/q, for some positive integers p,q}

• Positive rational numbers:
 INTERVAL NOTATION

[a,b] = {x | a ≤ x ≤ b}
[a,b) = {x | a ≤ x < b}
(a,b] = {x | a < x ≤ b}
(a,b) = {x | a < x < b}

closed interval [a,b]

open interval (a,b)
 UNIVERSAL SET AND EMPTY
SET
• The universal set U is the set containing everything currently
under consideration.
• Sometimes implicit Venn Diagram
• Sometimes explicitly stated.
• Contents depend on the context.
U
• The empty set is the set with no
elements. Symbolized by ∅. V aei
ou

John Venn (1834-1923)

Cambridge, UK
 SOME THINGS TO REMEMBER

{{1,2,3},a, {b,c}}
• Sets can be elements of sets.

{N,Z,Q,R}
• The empty set is different from a set containing the empty set.
∅ ≠{∅}
 SET EQUALITY
Definition: Two sets are equal if and only if they have the same
elements.
• Therefore if A and B are sets, then A and B are equal if and only if
.

{1,3,5} = {3, 5, 1}
• We write A = B if A and B are equal sets.

{1,5,5,5,3,3,1} = {1,3,5}
 SUBSETS
Definition: The set A is a subset of B, if and only if every

• The notation A ⊆ B is used to indicate that A is a subset of the set B.

element of A is also an element of B.

• A ⊆ B holds if and only if

Because a ∈ ∅ is always false, ∅ ⊆ S ,for every set S.
is true.

Because a ∈ S → a ∈ S, S ⊆ S, for every set S.

1.
2.
 SHOWING A SET IS OR IS NOT
A SUBSET OF ANOTHER SET
• Showing that A is a Subset of B: To show that A ⊆ B, show that if x
belongs to A, then x also belongs to B.

subset of B, A ⊈ B, find an element x ∈ A with x ∉ B. (Such an x is

• Showing that A is not a Subset of B: To show that A is not a

a counterexample to the claim that x ∈ A implies x ∈ B.)

1. The set of all computer science majors at your school is a subset of all
Examples:

students at your school.

2. The set of integers with squares less than 100 is not a subset of the set of
nonnegative integers.
 ANOTHER LOOK AT EQUALITY
OF SETS
• Recall that two sets A and B are equal, denoted by A = B, iff

• Using logical equivalences we have that A = B iff

• This is equivalent to
A⊆B and B⊆A
 PROPER SUBSETS
Definition: If A ⊆ B, but A ≠B, then we say A is a proper subset of
B, denoted by A ⊂ B. If A ⊂ B, then

is true.

U
Venn Diagram B
A
 SET CARDINALITY
Definition: If there are exactly n distinct elements in S where n is a
nonnegative integer, we say that S is finite. Otherwise it is infinite.
Definition: The cardinality of a finite set A, denoted by
|A|, is the number of (distinct) elements of A.
Examples:
1. |ø| = 0
2. Let S be the letters of the English alphabet. Then |S| = 26
3. |{1,2,3}| = 3
4. |{ø}| = 1
5. The set of integers is infinite.
 POWER SETS
Definition: The set of all subsets of a set A, denoted P(A), is
called the power set of A.
Example: If A = {a,b} then
P(A) = {ø, {a},{b},{a,b}}

2ⁿ. (In Chapters 5 and 6, we will discuss different ways to show

• If a set has n elements, then the cardinality of the power set is

this.)
 TUPLES
• The ordered n-tuple (a1,a2,…..,an) is the ordered collection that
has a1 as its first element and a2 as its second element and so
on until an as its last element.
• Two n-tuples are equal if and only if their corresponding
elements are equal.

• The ordered pairs (a,b) and (c,d) are equal if and only if a = c and
• 2-tuples are called ordered pairs.

b = d.
 Descartes
René

(1596-1650)
CARTESIAN PRODUCT
Definition: The Cartesian Product of two sets
A and B, denoted by A × B is the set of
ordered pairs (a,b) where a ∈ A and b ∈ B .

Example:
A = {a,b} B = {1,2,3}
A × B = {(a,1),(a,2),(a,3), (b,1),(b,2),(b,3)}

• Definition: A subset R of the Cartesian product

A × B is called a relation from the set A to the

Chapter 9. )
set B. (Relations will be covered in depth in
 CARTESIAN PRODUCT
Definition: The cartesian products of the sets A1,A2,……,An,
denoted by A1 × A2 × …… × An , is the set of ordered
(a1,a2,……,an) where ai belongs to Ai for i = 1, … n.
n-tuples

Example: What is A × B × C where A = {0,1}, B = {1,2} and C=

{0,1,2}
Solution: A × B × C = {(0,1,0), (0,1,1), (0,1,2),(0,2,0), (0,2,1),
(0,2,2),(1,1,0), (1,1,1), (1,1,2), (1,2,0), (1,2,1), (1,1,2)}
SET OPERATIONS
Section 2.2
 SECTION SUMMARY
• Set Operations
• Union
• Intersection
• Complementation
• Difference
• More on Set Cardinality
• Set Identities
• Proving Identities
• Membership Tables
 BOOLEAN ALGEBRA
• Propositional calculus and set theory are both instances of an
algebraic system called a Boolean Algebra. This is discussed in
CS2209.
• The operators in set theory are analogous to the corresponding
operator in propositional calculus.
• As always there must be a universal set U. All sets are assumed
to be subsets of U.
 UNION

denoted by A ∪ B, is the set:

• Definition: Let A and B be sets. The union of the sets A and B,

• Example: What is {1,2,3} ∪ {3, 4, 5}?

Venn Diagram for A ∪
Solution: {1,2,3,4,5} B
U
A B
 INTERSECTION
• Definition: The intersection of sets A and B, denoted by A ∩ B,
is

• Note if the intersection is empty, then A and B are said to be

disjoint.
• Example: What is {1,2,3} ∩ {3,4,5} ?
Venn Diagram for A ∩B
Solution: {3}
U
• Example:What is
{1,2,3} ∩ {4,5,6} ?
A B

Solution: ∅
 COMPLEMENT

respect to U), denoted by Ā is the set U - A

Definition: If A is a set, then the complement of the A (with

Ā = {x ∈ U | x ∉ A}
(The complement of A is sometimes denoted by Ac .)

complement of {x | x > 70}

Example: If U is the positive integers less than 100, what is the

Venn Diagram for

Complement U
Ā
A
 COMPLEMENT

respect to U), denoted by Ā is the set U - A

Definition: If A is a set, then the complement of the A (with

Ā = {x ∈ U | x ∉ A}
(The complement of A is sometimes denoted by Ac .)

complement of {x | x > 70}

Example: If U is the positive integers less than 100, what is the

Solution: {x | x ≤ 70}
Venn Diagram for
Complement U
Ā
A
 DIFFERENCE
• Definition: Let A and B be sets. The difference of A and B,
denoted by A – B, is the set containing the elements of A that are
not in B. The difference of A and B is also called the complement
of B with respect to A.
A – B = {x | x ∈ A  x ∉ B} = A ∩ Bc

U Venn Diagram for A − B

A
B
 THE CARDINALITY OF THE
UNION OF TWO SETS
|A ∪ B| = |A| + | B| - |A ∩ B|
• Inclusion-Exclusion
U
A B

Venn Diagram for A, B, A ∩ B, A

∪B
• Example: Let A be the math majors in your class and B be the CS majors.
To count the number of students who are either math majors or CS
majors, add the number of math majors and the number of CS majors,
and subtract the number of joint CS/math majors.
• We will return to this principle in Chapter 6 and Chapter 8 where we will
derive a formula for the cardinality of the union of n sets, where n is a
positive integer.
 REVIEW QUESTIONS
Example: U = {0,1,2,3,4,5,6,7,8,9,10} A = {1,2,3,4,5}, B ={4,5,6,7,8}
1. A ∪ B

2. A ∩ B

3. Ā

4.

5. A – B

6. B – A
 REVIEW QUESTIONS
Example: U = {0,1,2,3,4,5,6,7,8,9,10} A = {1,2,3,4,5}, B ={4,5,6,7,8}
1. A ∪ B

2. A ∩ B
Solution: {1,2,3,4,5,6,7,8}

Solution: {4,5}
3. Ā

4.
Solution: {0,6,7,8,9,10}

5. A – B
Solution: {0,1,2,3,9,10}

6. B – A
Solution: {1,2,3}

Solution: {6,7,8}
 SET IDENTITIES
• Identity laws

• Domination laws

• Idempotent laws

• Complementation law

Continued on next slide 

 SET IDENTITIES
• Commutative laws

• Associative laws

• Distributive laws

Continued on next slide 

 SET IDENTITIES
• De Morgan’s laws

• Absorption laws

• Complement laws
 PROVING SET IDENTITIES
• Prove that each set (side of the identity) is a subset of the
other.
• Use definitions and propositional logic
 PROOF OF SECOND DE
MORGAN LAW
Example: Prove that
Solution: We prove this identity by showing that:

1) and

2)

Continued on next slide 

 PROOF OF SECOND DE
MORGAN LAW
These steps show that:

Continued on next slide 

 PROOF OF SECOND DE
MORGAN LAW
These steps show that:
 GENERALIZED UNIONS AND
INTERSECTIONS
• Let A1, A2 ,…, An be an indexed collection of sets.
We define:

These are well defined, since union and intersection are

associative.
• For i = 1,2,…, let Ai = {i, i + 1, i + 2, ….}. Then,
 COMPUTER
REPRESENTATION OF SETS
• Assume the universal set U is finite (not larger than the memory size
of the computer being used)
• First specify an arbitrary ordering of the elements in U, for instance
a1, a2,…, an
• Represent a subset A of U with the bit string of length n, where the
ith bit is 1 if ai belongs to A and is 0 if ai does not belong to A

• Example: Let U ={1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, and the ordering of U

has the elements in the increasing order.

What bit strings represent the subset of all odd integers?

What bit string represents the set of all even integers?
What bit string represents the subset of integers not exceeding 5 in
U?

What is the bit string that represents the complement of a subset?

What is the bit string that represents the union of two subsets?
What is the bit string that represents the intersection of two
subsets?
FUNCTIONS
Section 2.3
 SECTION SUMMARY
• Definition of a Function.
• Domain, Codomain
• Image, Preimage
• Injection, Surjection, Bijection
• Inverse Function
• Function Composition
• Graphing Functions
• Floor, Ceiling, Factorial
• Partial functions
 FUNCTIONS

B, denoted f: A → B is an assignment of each element of A to exactly

Definition: Let A and B be nonempty sets. A function f from A to

one element of B. We write f(a) = b if b is the unique element of B

assigned by the function f to the element a of A.
• Functions are sometimes
called mappings or
Students Grade
s A
transformations. Carlota Rodriguez
B

Sandeep Patel C

Jalen Williams D

F
Kathy Scott
 FUNCTIONS
• A function f: A → B can also be defined as a subset of A×B (a
relation). This subset is restricted to be a relation where no two
elements of the relation have the same first element.
• Specifically, a function f from A to B contains one, and only one
ordered pair (a, b) for every element a∈ A.

and
 FUNCTIONS
Given a function f: A → B:
• We say f maps A to B or
f is a mapping from A to B.
• A is called the domain of f.
• B is called the codomain of f.
• If f(a) = b,

• a is called the preimage of b.

• then b is called the image of a under f.

• The range of f is the set of all images of points in A under f. We denote it by

f(A).
• Two functions are equal when they have the same domain, the same
codomain and map each element of the domain to the same element of the
codomain.
 REPRESENTING FUNCTIONS
• Functions may be specified in different ways:
• An explicit statement of the assignment.
Students and grades example.

f(x) = x + 1
• A formula.

• A computer program.

(covered in the next section and also in Chapter 5).

• A Java program that when given an integer n, produces the nth Fibonacci Number
 QUESTIONS
f(a) = ? A B
a
The image of d x
is ? b
The domain of f is y
?
The codomain of f
c

z
is ? d

The preimage of y
isf(A)
?
= ? preimage(s) of z is (are)
The
 QUESTIONS
f(a) = ? z A B
a
The image of d z x
is ? b
The domain of f isA y
?
The codomain of f isB
c

z
?
The preimage of y b
d

is ?
f(A) = ? {y,
z} preimage(s) of z is (are)
The {a,c,d}
 QUESTION ON FUNCTIONS
AND SETS
• If and S is a subset of A, then

A B
f {a,b,c,} a
x
is ? b
f {c,d} is ? y
c

d z
 QUESTION ON FUNCTIONS
AND SETS
• If and S is a subset of A, then

A B
f {a,b,c,} {y,z} a
x
is ? b
f {c,d} is ? {z y
} c

d z
 INJECTIONS
Definition: A function f is said to be one-to-one , or injective, if
and only if f(a) = f(b) implies that a = b for all a and b in the
domain of f. A function is said to be an injection if it is one-to-
one.
A B
a x

v
b
y
c
z
d

w
 SURJECTIONS
Definition: A function f from A to B is called onto or surjective, if
and only if for every element there is an element
with . A function f is called a surjection if it is onto.

A B
a x

b
y
c
z
d
 BIJECTIONS
Definition: A function f is a one-to-one correspondence, or a
bijection, if it is both one-to-one and onto (surjective and
injective).

A
a
B
x

b
y
c

d z

w
SHOWING THAT F IS ONE-TO-
ONE OR ONTO
 SHOWING THAT F IS ONE-TO-
ONE OR ONTO
Example 1: Let f be the function from {a,b,c,d} to {1,2,3}
defined by f(a) = 3, f(b) = 2, f(c) = 1, and f(d) = 3. Is f an onto
function?

Example 2: Is the function f(x) = x2 from the set of integers

onto?
 SHOWING THAT F IS ONE-TO-
ONE OR ONTO
Example 1: Let f be the function from {a,b,c,d} to {1,2,3}
defined by f(a) = 3, f(b) = 2, f(c) = 1, and f(d) = 3. Is f an onto
function?
Solution: Yes, f is onto since all three elements of the codomain
are images of elements in the domain. If the codomain were
changed to {1,2,3,4}, f would not be onto.
Example 2: Is the function f(x) = x2 from the set of integers
onto?

= −1, for example.

Solution: No, f is not onto because there is no integer x with x2
 INVERSE FUNCTIONS
Definition: Let f be a bijection from A to B. Then the inverse of f,
denoted , is the function from B to A defined as
No inverse exists unless f is a bijection. Why?
 INVERSE FUNCTIONS
A f
B A B
a V V
a

b b
W W
c c

d X X
d

Y Y
 QUESTIONS
Example 1: Let f be the function from {a,b,c} to {1,2,3} such
that f(a) = 2, f(b) = 3, and f(c) = 1. Is f invertible and if so what is
its inverse?
 QUESTIONS
Example 1: Let f be the function from {a,b,c} to {1,2,3} such
that f(a) = 2, f(b) = 3, and f(c) = 1. Is f invertible and if so what is
its inverse?

Solution: The function f is invertible

because it is a one-to-one correspondence.
The inverse function f-1 reverses the
correspondence given by f, so f-1 (1) = c, f-1
(2) = a, and f-1 (3) = b.
 QUESTIONS
Example 2: Let f: Z  Z be such that f(x) = x + 1. Is f invertible,
and if so, what is its inverse?
 QUESTIONS
Example 2: Let f: Z  Z be such that f(x) = x + 1. Is f invertible,
and if so, what is its inverse?

Solution: The function f is invertible because it is

a one-to-one correspondence. The inverse

= y – 1.
function f-1 reverses the correspondence so f-1 (y)
 QUESTIONS
Example 3: Let f: R → R be such that . Is f invertible,
and if so, what is its inverse?
 QUESTIONS
Example 3: Let f: R → R be such that . Is f invertible,
and if so, what is its inverse?

Solution: The function f is not invertible

because it is not one-to-one .
 COMPOSITION
• Definition: Let f: B → C, g: A → B. The composition of f with g,
denoted is the function from A to C defined by
 COMPOSITION
g f
A B C A C
V a
a h h
b i b
W i
c
c
X j
d
d j
Y
 COMPOSITION
Example 1: If and , then

and
 COMPOSITION QUESTIONS
Example 2: Let g be the function from the set {a,b,c} to itself

f(a) = 3, f(b) =
such that g(a) = b, g(b) = c, and g(c) = a. Let f be the function

2, and f(c) = 1.
from the set {a,b,c} to the set {1,2,3} such that

What is the composition of f and g, and what is the composition

of g and f.
 COMPOSITION QUESTIONS
Example 2: Let g be the function from the set {a,b,c} to itself

f(a) = 3, f(b) =
such that g(a) = b, g(b) = c, and g(c) = a. Let f be the function

2, and f(c) = 1.
from the set {a,b,c} to the set {1,2,3} such that

What is the composition of f and g, and what is the composition

Solution: The composition f∘g is defined by

of g and f.

f∘g (a)= f(g(a)) = f(b) = 2.

f∘g (b)= f(g(b)) = f(c) = 1.
f∘g (c)= f(g(c)) = f(a) = 3.
Note that g∘f is not defined, because the range of f is not a subset of the
domain of g.
 COMPOSITION QUESTIONS
Example 2: Let f and g be functions from the set of integers to
the set of integers defined by f(x) = 2x + 3 and g(x) = 3x + 2.
What is the composition of f and g, and also the composition of g
and f ?
 COMPOSITION QUESTIONS
Example 2: Let f and g be functions from the set of integers to
the set of integers defined by f(x) = 2x + 3 and g(x) = 3x + 2.
What is the composition of f and g, and also the composition of g
and f ?

f∘g (x)= f(g(x)) = f(3x + 2) = 2(3x + 2) + 3 = 6x + 7

Solution:

g∘f (x)= g(f(x)) = g(2x + 3) = 3(2x + 3) + 2 = 6x + 11

 GRAPHS OF FUNCTIONS

function f is the set of ordered pairs {(a,b) | a ∈A and f(a) = b}.

• Let f be a function from the set A to the set B. The graph of the

Graph of f(n) = 2n + 1 Graph of f(x) = x2

from Z to Z from Z to Z
 SOME IMPORTANT
FUNCTIONS
• The floor function, denoted

is the largest integer less than or equal to x.

• The ceiling function, denoted

is the smallest integer greater than or equal to x

Example:
 FLOOR AND CEILING
FUNCTIONS

Graph of (a) Floor and (b) Ceiling

Functions
FLOOR AND CEILING
FUNCTIONS
 COMPUTER APPLICATIONS OF
FLOOR AND CEILING FUNCTIONS
• Example: Data stored on a computer or transmitted over a
network are represented as a string of bytes. Each byte consists
of 8 bits. How many bytes are required to encode 100 bits of
data?
• Example: In asynchronous transfer mode (ATM)
(a communication protocol used in backbone networks), data are
organized into cells of 53 bytes. How many ATM cells can be
transmitted in 1 minute over a connection that transmits data at
a rate of 500 kilobits per second?
 PROVING PROPERTIES OF
FUNCTIONS

⌊2x⌋= ⌊x⌋ + ⌊x + ½⌋
Example: Prove that x is a real number, then
 PROVING PROPERTIES OF
FUNCTIONS
⌊2x⌋= ⌊x⌋ + ⌊x + 1/2⌋
Example: Prove that x is a real number, then

Solution: Let x = n + ε, where n is an integer and 0 ≤ ε< 1.

Case 1: ε<½
• 2x = 2n + 2ε and ⌊2x⌋ = 2n, since 0 ≤ 2ε< 1.
• ⌊x + 1/2⌋ = n, since x + ½ = n + (1/2 + ε ) and 0 ≤ ½ +ε < 1.
• Hence, ⌊2x⌋ = 2n and ⌊x⌋ + ⌊x + 1/2⌋ = n + n = 2n.
Case 2: ε≥½
• 2x = 2n + 2ε = (2n + 1) +(2ε − 1) and ⌊2x⌋ =2n + 1, since 0 ≤ 2 ε -
1< 1.
• ⌊x + 1/2⌋ = ⌊ n + (1/2 + ε)⌋ = ⌊ n + 1 + (ε – 1/2)⌋ = n + 1 since 0 ≤ ε – 1/2< 1.
• Hence, ⌊2x⌋ = 2n + 1 and ⌊x⌋ + ⌊x + 1/2⌋ = n + (n + 1) = 2n + 1.
 FACTORIAL FUNCTION
Definition: f: N → Z+ , denoted by f(n) = n! is the product of the
first n positive integers when n is a nonnegative integer.

f(n) = 1 ∙ 2 ∙∙∙ (n – 1) ∙ n, f(0) = 0! = 1

Examples:
f(1) = 1! = 1

f(2) = 2! = 1 ∙ 2 = 2

f(6) = 6! = 1 ∙ 2 ∙ 3∙ 4∙ 5 ∙ 6 = 720
Stirling’s Formula:
f(20) = 2,432,902,008,176,640,000.
 PARTIAL FUNCTIONS
Definition: A partial function f from a set A to a set B is an
assignment to each element a in a subset of A, called the domain
of definition of f, of a unique element b in B.
• The sets A and B are called the domain and codomain of f, respectively.
• We day that f is undefined for elements in A that are not in the domain of
definition of f.
• When the domain of definition of f equals A, we say that f is a total
function.

Example: f: N → R where f(n) = √n is a partial function from Z to

R where the domain of definition is the set of nonnegative
integers. Note that f is undefined for negative integers.
 SAMPLE PROBLEM
•Use a Venn diagram to determine which relationship, Í, =, Ê, is true for the pair of sets.
• A È B, A È (B - A).
• A È (B Ç C), (A È B) Ç C.
• (A - B) È (A - C), A - (B Ç C).
 CONCEPT
• AÈB=A
• B is a subset of A

• AÇB=A
• A is a subset of B

• If the result is neither A nor B, then A and B are not equal and not a subset
 A È B, A È (B - A).
(A È B) = (A È (B - A))
= (A È (B Ç Ac)) 2nd form of difference
= (A È B) Ç (A È Ac) Distributive law
= (A È B) Ç U Complement laws
= (A È B) Identity laws

Therefore: equal
 A È (B Ç C), (A È B) Ç C
A È (B Ç C) = (A È B) Ç C
= (A Ç C) È (B Ç C) Distributive law

Not equal
 A È (B Ç C), (A È B) Ç C
Test if one set is a subset of the another set

A È (B Ç C) È (A È B) Ç C
A È (B Ç C) È (A Ç C) È (B Ç C) Distributive law
[A È (A Ç C) ]È [(B Ç C) È (B Ç C)] Idempotent, Commutative and Associative law
A È (B Ç C) Idempotent Law

Therefore
(A È B) Ç C Í A È (B Ç C)
 (A - B) È (A - C), A - (B Ç C).
• (A - B) È (A - C) = A - (B Ç C)
• (A Ç Bc) È (A Ç Cc) 2ND Form of Difference
• A Ç (Bc È Cc) Distributive Law
• A Ç (B Ç C) c De Morgan’s Law
• A - (B Ç C) 1st form of Differnce

• Therefore, equal
THE END 