Class Notes for Discrete Math I (Rosen)
2.2: SET OPERATIONS
Set Operations: Let A and B be sets with U the universal set.
 Union: AB = {x | xA  xB}.
 Intersection: AB = {x | xA  xB}.
 Difference (or Complement of A with respect to B):
A  B = {x | xA  xB}.
 Complement: A  U  A  x | x  A .
Two sets are disjoint if their intersection is empty.
Example 1: Let A = {1,2,3} and B = {3,4,5}. Determine the
following.
(a) AB
(d) BA
(b) AB
(e) A  (A  B)
(c) AB
Example 2: Let A = [1,3] and B = (2,5]. Determine the following.
(a) AB
(b) AB
(c) AB
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Class Notes for Discrete Math I (Rosen)
Example 3: Let
A = {n+ | n is even} and B = { n+ | n is prime}
with U = + the universal set. Determine the following.
(a) A
(c) A  B
(b)
(d) A  B
 A
SET IDENTITIES: Let A, B and C be sets with U the universal set.
Complementation law:  A   A
Identity laws:
A   = A,
AU=A
Domination laws:
A  U = U,
A=
Complement laws:
A A U ,
A A  
Idempotent laws:
A  A = A,
AA=A
Commutative laws:
A  B = B  A,
AB=BA
Associative laws:
A  (B  C) = (A  B)  C,
A  (B  C) = (A  B)  C
Distributive laws:
A  (B  C) = (A  B)  (A  C),
A  (B  C) = (A  B)  (A  C)
Absorption laws:
A  (A  B) = A, A  (A  B) = A
DeMorgan’s laws:
A B  A B ,
A B  A B
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Class Notes for Discrete Math I (Rosen)
Example 4: Prove DeMorgan’s Law A  B  A  B .
Defintion: Let {A1, A2, …, An} be a collection of sets.
 The union U of {A1, A2, …, An} is the set that contains
elements that are members of at least one of the sets in the
collection. That is,
n
U
A1
A2
An
Ai
i 1
 The intersection I of {A1, A2, …, An} is the set that contains
elements that are members of all of the sets in the
collection. That is,
n
I
A1
A2
An
Ai
i 1
Example 5: If Ai = {0, 1, 2, …, i} for all positive integers i, then
determine
30
Ai
(a)
i 1
30
Ai
(b)
i 1
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Class Notes for Discrete Math I (Rosen)
SECTION 2.2 – SET OPERATIONS
Set Operations: Let A and B be sets with U the universal set.
 Union: AB = {x | xA  xB}.
 Intersection: AB = {x | xA  xB}.
 Difference (or Complement of A with respect to B): AB = {x | xA  xB}.
 Complement: A  U  A  x | x  A .
Venn Diagrams: Let A and B be sets with U the universal set.
AB
AB
U
A
B
A
AB
B
A
U
A
U
B
U
A
Two sets are disjoint if their intersection is empty. I.e., A and B are disjoint if AB =.
Example 1: Let A = {1,2,3} and B = {3,4,5}. Determine the following.
(a) AB
(d) BA
(b) AB
(e) A  (A  B)
(c) AB
Example 2: Let A = [1,3] and B = (2,5]. Determine the following.
(a) AB
(b) AB
(c) AB
Example 3: Let A = {n+ | n is even} and B = { n+ | n is prime} with U = + the
universal set. Determine the following.
(a) A
(c) A  B
(b)
(d) A  B
 A
SET IDENTITIES
Set Identities: Let A, B and C be sets with U the universal set.
Complementation law:  A   A
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Class Notes for Discrete Math I (Rosen)
Identity laws:
Domination laws:
Complement laws:
Idempotent laws:
Commutative laws:
Associative laws:
Distributive laws:
Absorption laws:
DeMorgan’s laws:
A   = A,
A  U = U,
AU=A
A=
A A U ,
A A  
A  A = A,
A  B = B  A,
A  (B  C) = (A  B)  C,
A  (B  C) = (A  B)  (A  C),
A  (A  B) = A,
A B  A  B ,
AA=A
AB=BA
A  (B  C) = (A  B)  C
A  (B  C) = (A  B)  (A  C)
A  (A  B) = A
A B  A  B
Compare with the Logical Equivalences on pg 24 of the textbook.
Example 4: Prove DeMorgan’s Law A  B  A  B .
Proof:


A B  x | x A B
  x | x  A  B
  x | x  A  B
  x |   x  A  x  B 
  x | x  A  x  B
  x | x  A  x  B
  x | x  A  x  B
 AB
Or, using Venn diagrams, we have
A B
A
A B

B
A
B
A B
A
B
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Class Notes for Discrete Math I (Rosen)
Venn Diagrams are not considered “proofs”, but merely a guide in understanding a given
identity.
GENERALIZED UNIONS AND INTERSECTIONS
Defintion: Let {A1, A2, …, An} be a collection of sets.
 The union U of {A1, A2, …, An} is the set that contains elements that are members of at
least one of the sets in the collection. That is,
n
U
A1
A2
An
Ai
i 1
 The intersection I of {A1, A2, …, An} is the set that contains elements that are members of
all of the sets in the collection. That is,
n
I
A1
A2
An
Ai
i 1
Example 5: If Ai = {0, 1, 2, …, i} for all positive integers i, then determine
30
(a)
30
Ai , (b)
i 1
Ai
i 1
92