Definition 1
The Cartesian product (or cross product) of A and B, denoted by
A × B, is the set
A × B = {(a, b) | a ∈ A and b ∈ B}
1. the elements (a, b) of A × B are ordered pairs
2. for pairs (a, b), (c, d) we have
(a, b) = (c, d) ⇐⇒ a = c and b = d
Definition 2
The n-fold product of sets A1 , A2 , . . . , An is the set of n-tuples
A1 × A2 × . . . × An = {(a1 , a2 , . . . , an ) | ai ∈ Ai for all 1 ≤ i ≤ n}
()
October 30, 2007
1 / 12
Definition 1
The Cartesian product (or cross product) of A and B, denoted by
A × B, is the set
A × B = {(a, b) | a ∈ A and b ∈ B}
1. the elements (a, b) of A × B are ordered pairs
2. for pairs (a, b), (c, d) we have
(a, b) = (c, d) ⇐⇒ a = c and b = d
Definition 2
The n-fold product of sets A1 , A2 , . . . , An is the set of n-tuples
A1 × A2 × . . . × An = {(a1 , a2 , . . . , an ) | ai ∈ Ai for all 1 ≤ i ≤ n}
()
October 30, 2007
1 / 12
Definition 1
The Cartesian product (or cross product) of A and B, denoted by
A × B, is the set
A × B = {(a, b) | a ∈ A and b ∈ B}
1. the elements (a, b) of A × B are ordered pairs
2. for pairs (a, b), (c, d) we have
(a, b) = (c, d) ⇐⇒ a = c and b = d
Definition 2
The n-fold product of sets A1 , A2 , . . . , An is the set of n-tuples
A1 × A2 × . . . × An = {(a1 , a2 , . . . , an ) | ai ∈ Ai for all 1 ≤ i ≤ n}
()
October 30, 2007
1 / 12
Definition 1
A × B = {(a, b) | a ∈ A and b ∈ B}
A = {2, 3, 4}
a) A×B = {(2, 4), (2, 5), (3, 4), (3, 5), (4, 4), (4, 5)}
B = {4, 5}
b) B ×A = {(4, 2), (4, 3), (4, 4), (5, 2), (5, 3), (5, 4)}
O
B
A×B
◦ ◦ ◦
5
4
◦
◦
◦
/
2
()
3
4
A
October 30, 2007
2 / 12
Definition 1
A × B = {(a, b) | a ∈ A and b ∈ B}
A = {2, 3, 4}
a) A×B = {(2, 4), (2, 5), (3, 4), (3, 5), (4, 4), (4, 5)}
B = {4, 5}
b) B ×A = {(4, 2), (4, 3), (4, 4), (5, 2), (5, 3), (5, 4)}
O
B
A×B
◦ ◦ ◦
5
4
◦
◦
◦
/
2
()
3
4
A
October 30, 2007
2 / 12
Definition 1
A × B = {(a, b) | a ∈ A and b ∈ B}
A = {2, 3, 4}
a) A×B = {(2, 4), (2, 5), (3, 4), (3, 5), (4, 4), (4, 5)}
B = {4, 5}
b) B ×A = {(4, 2), (4, 3), (4, 4), (5, 2), (5, 3), (5, 4)}
O
B
O
A×B
A
◦ ◦ ◦
B ×A
5
4
◦
◦
◦
4
3
2
◦ ◦
◦ ◦
◦ ◦
/
2
()
3
4
A
/
4
5
B
October 30, 2007
2 / 12
B = {4, 5}
a) B 2 = B × B = {(4, 4), (4, 5), (5, 4), (5, 5)}
b) B 3 = B × B × B = {(4, 4, 4), (4, 4, 5), (4, 5, 4), (4, 5, 5),
(5, 4, 4), (5, 4, 5), (5, 5, 4), (5, 5, 5)}
B
5
4
O
B ×B
◦ ◦
◦
◦
/
4
()
5
B
October 30, 2007
3 / 12
B = {4, 5}
a) B 2 = B × B = {(4, 4), (4, 5), (5, 4), (5, 5)}
b) B 3 = B × B × B = {(4, 4, 4), (4, 4, 5), (4, 5, 4), (4, 5, 5),
(5, 4, 4), (5, 4, 5), (5, 5, 4), (5, 5, 5)}
B
5
4
O
B ×B
◦ ◦
◦
◦
/
4
()
5
B
October 30, 2007
3 / 12
B = {4, 5}
a) B 2 = B × B = {(4, 4), (4, 5), (5, 4), (5, 5)}
b) B 3 = B × B × B = {(4, 4, 4), (4, 4, 5), (4, 5, 4), (4, 5, 5),
(5, 4, 4), (5, 4, 5), (5, 5, 4), (5, 5, 5)}
B
◦ ◦
◦
B ×B ×B
◦ ◦
◦ ◦
O
◦
5
t:
tt
t
tt B
44tttt
4 ttt
tt4 5
t
t
tt
4
tt
t
t
t
tt
tt
tt
4
4
()
5
/
B
October 30, 2007
3 / 12
g (4, x, ♥)
ggggg
(4, x) WWWW (4, x, ♠)
WW
oo
o
o
oo
(4, x, ♣)
(4) OO
OOO
O

g (4, y , ♥)
ggggg



(4, y ) WWWW (4, y , ♠)

WW


(4, y , ♣)

()??
??
(, x, ♥)
??
ggg
g
g
g
??
(, x) WWWW (, x, ♠)
??
WW
??
oo
o
o
oo
(, x, ♣)
() OO
OOO
O
g (, y , ♥)
ggggg
(, y ) WWWW (, y , ♠)
WW
{4, } × {x, y } × {♥, ♠, ♣}
Tree Diagram
(, y , ♣)
()
October 30, 2007
4 / 12
Definition 3
A (binary) relation from A to B is a subset of A × B.
A (binary) relation on A is a subset of A × A.
B
A = {2, 3, 4} and B = {4, 5}
a)
b)
c)
d)
R1
R2
R3
R4
= {(2, 4), (3, 5)}
= {(2, 4), (3, 4), (4, 4)}
= {(2, 4), (2, 5), (4, 4), (4, 5)}
=∅
O
R1
◦
5
4
◦
/
2
()
3
4
October 30, 2007
A
5 / 12
Definition 3
A (binary) relation from A to B is a subset of A × B.
A (binary) relation on A is a subset of A × A.
B
A = {2, 3, 4} and B = {4, 5}
a)
b)
c)
d)
R1
R2
R3
R4
= {(2, 4), (3, 5)}
= {(2, 4), (3, 4), (4, 4)}
= {(2, 4), (2, 5), (4, 4), (4, 5)}
=∅
O
R1
◦
5
4
◦
/
2
()
3
4
October 30, 2007
A
5 / 12
Definition 3
A (binary) relation from A to B is a subset of A × B.
A (binary) relation on A is a subset of A × A.
B
O
R2
A = {2, 3, 4} and B = {4, 5}
a)
b)
c)
d)
R1
R2
R3
R4
= {(2, 4), (3, 5)}
= {(2, 4), (3, 4), (4, 4)}
= {(2, 4), (2, 5), (4, 4), (4, 5)}
=∅
5
4
◦
◦ ◦
/
2
()
3
4
October 30, 2007
A
5 / 12
Definition 3
A (binary) relation from A to B is a subset of A × B.
A (binary) relation on A is a subset of A × A.
B
O
R3
A = {2, 3, 4} and B = {4, 5}
a)
b)
c)
d)
R1
R2
R3
R4
= {(2, 4), (3, 5)}
= {(2, 4), (3, 4), (4, 4)}
= {(2, 4), (2, 5), (4, 4), (4, 5)}
=∅
5
◦
◦
4
◦
◦
/
2
()
3
4
October 30, 2007
A
5 / 12
Definition 3
A (binary) relation from A to B is a subset of A × B.
A (binary) relation on A is a subset of A × A.
B
A = {2, 3, 4} and B = {4, 5}
a)
b)
c)
d)
R1
R2
R3
R4
= {(2, 4), (3, 5)}
= {(2, 4), (3, 4), (4, 4)}
= {(2, 4), (2, 5), (4, 4), (4, 5)}
=∅
O
R4
5
4
/
2
()
3
4
October 30, 2007
A
5 / 12
Relation R = {(x, y ) ∈ Z × Z | 1 ≤ x ≤ y ≤ 4} =
= {(1, 1), (1, 2), (1, 3), (1, 4), (2, 2), (2, 3), (2, 4), (3, 3), (3, 4), (4, 4)}
O
R
4
◦
◦ ◦
3
◦
◦ ◦
2
◦
◦
◦
1
Notation
(x, y ) ∈ R
◦
m
xRy
/
1
()
2
3
(think of R as ≤)
4
October 30, 2007
6 / 12
Relation R = {(x, y ) ∈ Z × Z | 1 ≤ x ≤ y ≤ 4} =
= {(1, 1), (1, 2), (1, 3), (1, 4), (2, 2), (2, 3), (2, 4), (3, 3), (3, 4), (4, 4)}
O
R
4
◦
◦ ◦
3
◦
◦ ◦
2
◦
◦
◦
1
Notation
(x, y ) ∈ R
◦
m
xRy
/
1
()
2
3
(think of R as ≤)
4
October 30, 2007
6 / 12
Relation R = {(x, y ) ∈ Z × Z | 1 ≤ x ≤ y ≤ 4} =
= {(1, 1), (1, 2), (1, 3), (1, 4), (2, 2), (2, 3), (2, 4), (3, 3), (3, 4), (4, 4)}
O
R
4
◦
◦ ◦
3
◦
◦ ◦
2
◦
◦
◦
1
Notation
(x, y ) ∈ R
◦
m
xRy
/
1
()
2
3
(think of R as ≤)
4
October 30, 2007
6 / 12
{Justin, Joey , Kevin, Nick} × {Britney , Christina, Jessica, Kelly , Sarah}
dddddddoBritney
ddddddd ooozozozz
d
d
d
Z
Z
Justin:DO:DODOZOZZZZZoZoZoZoZ zz
::DDOOoOoo zzZZZZZd
O z d
od:od:D:odDdDDddOzOdzOdzOdOddododozozozChristina
o
o
d
d
Joey DOZDOZOZZZ:zZ:z:zZzDZDoZDoZoZooOzOzOzOO
DDOzzOOoo:o: DDzZzZZZZZO
zozoDoDoDdOdOdO:dzO:dzOdzDdDdDDddododoJessica
z
zoo ddd D z : O oo
KevinZOdOZdOZOZZzZzZzDZzDDZoDZoo:Z:oZ:OoZOODODODOD
zOzOoOoo DD ::ZZZdZ
zozozodododdOdOdOdOOddDDdDd:Dd::d Kelly
dzo
d
O
Nick ZdZZZZZZZZZZZOOODOD:OD:D:
ZZZZZO
Sarah
()
October 30, 2007
7 / 12
Who dated whom? {(Ju, Br ), (Ju, Je), (Jo, Ke), (Jo, Sa),
(Ke, Br ), (Ke, Ch), (Ni, Ch), (Ni, Ke), (Ni, Sa)}
dd
ddddddd zzzzBritney
ddddddd
z
d
z
zzzz
JustinOOOO
OOOO zzzzzzz
OOzzzOzz ooozozzChristina
zz OO
z
z
z
z
O
Joey DDDODO zzzz oooooOzozOzzOzzOzz
z
z
O
D
O
o
O
DDDzzDzOzOoo zzzz
zzoDoDDo OOOOzzz
Jessica
z
z
z
D
zozzozo DDDDDzzzzzOOOO
z
D
z
D
D
z
Kevin
O
OO
zz DD
zzzzz DDDDDDD ddddOKelly
z
z
d
D
z
d
d
zzzzdzdddddddd DDDDD
Nick dZZZZZZZZZZZ ZZDDDDDDDD
Z ZZZ
Sarah
()
October 30, 2007
8 / 12
Who is dating whom? {(Ju, Je), (Ke, Br ), (Ni, Ch), (Ni, Ke)}
Britney
Justin
Christina
Joey
Jessica
Kevin
Kelly
Nick
Sarah
()
October 30, 2007
9 / 12
Theorem 4
For any set A, we have A × ∅ = ∅ ( and ∅ × A = ∅)
Proof. If (a, b) ∈ A × ∅, then a ∈ A and b ∈ ∅, impossible.
Theorem 5
For any sets A, B, C
a) A × (B ∩ C ) = (A × B) ∩ (A × C )
b) A × (B ∪ C ) = (A × B) ∪ (A × C )
c) (A ∩ B) × C = (A × C ) ∩ (B × C )
d) (A ∪ B) × C = (A × C ) ∪ (B × C )
Proof. a) (a, b) ∈ A × (B ∩ C ) ⇐⇒ a ∈ A and b ∈ B ∩ C ⇐⇒
a ∈ A and b ∈ B and b ∈ C ⇐⇒ (a, b) ∈ A × B and
(a, b) ∈ A × C ⇐⇒ (a, b) ∈ (A × B) ∩ (A × C )
()
October 30, 2007
10 / 12
Theorem 4
For any set A, we have A × ∅ = ∅ ( and ∅ × A = ∅)
Proof. If (a, b) ∈ A × ∅, then a ∈ A and b ∈ ∅, impossible.
Theorem 5
For any sets A, B, C
a) A × (B ∩ C ) = (A × B) ∩ (A × C )
b) A × (B ∪ C ) = (A × B) ∪ (A × C )
c) (A ∩ B) × C = (A × C ) ∩ (B × C )
d) (A ∪ B) × C = (A × C ) ∪ (B × C )
Proof. a) (a, b) ∈ A × (B ∩ C ) ⇐⇒ a ∈ A and b ∈ B ∩ C ⇐⇒
a ∈ A and b ∈ B and b ∈ C ⇐⇒ (a, b) ∈ A × B and
(a, b) ∈ A × C ⇐⇒ (a, b) ∈ (A × B) ∩ (A × C )
()
October 30, 2007
10 / 12
Observation 6
For any two sets A, B, the number of elements in A × B is
|A × B| = |A| · |B|
Hence there are exactly |P(A × B)| = 2|A×B| = 2|A|·|B| different
relations from A to B.
()
October 30, 2007
11 / 12
Observation 6
For any two sets A, B, the number of elements in A × B is
|A × B| = |A| · |B|
Hence there are exactly |P(A × B)| = 2|A×B| = 2|A|·|B| different
relations from A to B.
()
October 30, 2007
11 / 12
Exercises:
5.1.7 - If A = {1, 2, 3, 4, 5} and B = {w , x, y , z}, how many
elements are there in P(A × B).
Answer: 220 = 1, 048, 576
5.1.3 - For A = {1, 2, 3} and B = {2, 4, 5}
a) |A × B| =?
Answer: 9
b) # of relations from A to B ?
Answer: 29 = 512
c) # of relations on A ?
Answer: 29 = 512
d) # of relations from A to B
that contain (1, 2) and (1, 5) ?
Answer: 27 = 128
e) # of relations from A to B
that contain exactly five ordered pairs ?
Answer: 95 = 126
f) # of relations on A that
contain at least seven elements ? Answer: 97 + 98 + 99 = 121
()
October 30, 2007
12 / 12
Exercises:
5.1.7 - If A = {1, 2, 3, 4, 5} and B = {w , x, y , z}, how many
Answer: 220 = 1, 048, 576
elements are there in P(A × B).
5.1.3 - For A = {1, 2, 3} and B = {2, 4, 5}
a) |A × B| =?
Answer: 9
b) # of relations from A to B ?
Answer: 29 = 512
c) # of relations on A ?
Answer: 29 = 512
d) # of relations from A to B
that contain (1, 2) and (1, 5) ?
Answer: 27 = 128
e) # of relations from A to B
that contain exactly five ordered pairs ?
Answer: 95 = 126
f) # of relations on A that
contain at least seven elements ? Answer: 97 + 98 + 99 = 121
()
October 30, 2007
12 / 12
Exercises:
5.1.7 - If A = {1, 2, 3, 4, 5} and B = {w , x, y , z}, how many
Answer: 220 = 1, 048, 576
elements are there in P(A × B).
5.1.3 - For A = {1, 2, 3} and B = {2, 4, 5}
a) |A × B| =?
Answer: 9
b) # of relations from A to B ?
Answer: 29 = 512
c) # of relations on A ?
Answer: 29 = 512
d) # of relations from A to B
that contain (1, 2) and (1, 5) ?
Answer: 27 = 128
e) # of relations from A to B
that contain exactly five ordered pairs ?
Answer: 95 = 126
f) # of relations on A that
contain at least seven elements ? Answer: 97 + 98 + 99 = 121
()
October 30, 2007
12 / 12
Exercises:
5.1.7 - If A = {1, 2, 3, 4, 5} and B = {w , x, y , z}, how many
Answer: 220 = 1, 048, 576
elements are there in P(A × B).
5.1.3 - For A = {1, 2, 3} and B = {2, 4, 5}
a) |A × B| =?
Answer: 9
b) # of relations from A to B ?
Answer: 29 = 512
c) # of relations on A ?
Answer: 29 = 512
d) # of relations from A to B
that contain (1, 2) and (1, 5) ?
Answer: 27 = 128
e) # of relations from A to B
that contain exactly five ordered pairs ?
Answer: 95 = 126
f) # of relations on A that
contain at least seven elements ? Answer: 97 + 98 + 99 = 121
()
October 30, 2007
12 / 12
Exercises:
5.1.7 - If A = {1, 2, 3, 4, 5} and B = {w , x, y , z}, how many
Answer: 220 = 1, 048, 576
elements are there in P(A × B).
5.1.3 - For A = {1, 2, 3} and B = {2, 4, 5}
a) |A × B| =?
Answer: 9
b) # of relations from A to B ?
Answer: 29 = 512
c) # of relations on A ?
Answer: 29 = 512
d) # of relations from A to B
that contain (1, 2) and (1, 5) ?
Answer: 27 = 128
e) # of relations from A to B
that contain exactly five ordered pairs ?
Answer: 95 = 126
f) # of relations on A that
contain at least seven elements ? Answer: 97 + 98 + 99 = 121
()
October 30, 2007
12 / 12
Exercises:
5.1.7 - If A = {1, 2, 3, 4, 5} and B = {w , x, y , z}, how many
Answer: 220 = 1, 048, 576
elements are there in P(A × B).
5.1.3 - For A = {1, 2, 3} and B = {2, 4, 5}
a) |A × B| =?
Answer: 9
b) # of relations from A to B ?
Answer: 29 = 512
c) # of relations on A ?
Answer: 29 = 512
d) # of relations from A to B
that contain (1, 2) and (1, 5) ?
Answer: 27 = 128
e) # of relations from A to B
that contain exactly five ordered pairs ?
Answer: 95 = 126
f) # of relations on A that
contain at least seven elements ? Answer: 97 + 98 + 99 = 121
()
October 30, 2007
12 / 12
Exercises:
5.1.7 - If A = {1, 2, 3, 4, 5} and B = {w , x, y , z}, how many
Answer: 220 = 1, 048, 576
elements are there in P(A × B).
5.1.3 - For A = {1, 2, 3} and B = {2, 4, 5}
a) |A × B| =?
Answer: 9
b) # of relations from A to B ?
Answer: 29 = 512
Answer: 29 = 512
c) # of relations on A ?
d) # of relations from A to B
that contain (1, 2) and (1, 5) ?
Answer: 27 = 128
e) # of relations from A to B
that contain exactly five ordered pairs ?
Answer: 95 = 126
f) # of relations on A that
contain at least seven elements ? Answer: 97 + 98 + 99 = 121
()
October 30, 2007
12 / 12
Exercises:
5.1.7 - If A = {1, 2, 3, 4, 5} and B = {w , x, y , z}, how many
Answer: 220 = 1, 048, 576
elements are there in P(A × B).
5.1.3 - For A = {1, 2, 3} and B = {2, 4, 5}
a) |A × B| =?
Answer: 9
b) # of relations from A to B ?
Answer: 29 = 512
Answer: 29 = 512
c) # of relations on A ?
d) # of relations from A to B
that contain (1, 2) and (1, 5) ?
Answer: 27 = 128
e) # of relations from A to B
that contain exactly five ordered pairs ?
Answer: 95 = 126
f) # of relations on A that
contain at least seven elements ? Answer: 97 + 98 + 99 = 121
()
October 30, 2007
12 / 12
Exercises:
5.1.7 - If A = {1, 2, 3, 4, 5} and B = {w , x, y , z}, how many
Answer: 220 = 1, 048, 576
elements are there in P(A × B).
5.1.3 - For A = {1, 2, 3} and B = {2, 4, 5}
a) |A × B| =?
Answer: 9
b) # of relations from A to B ?
Answer: 29 = 512
Answer: 29 = 512
c) # of relations on A ?
d) # of relations from A to B
Answer: 27 = 128
that contain (1, 2) and (1, 5) ?
e) # of relations from A to B
that contain exactly five ordered pairs ?
Answer: 95 = 126
f) # of relations on A that
contain at least seven elements ? Answer: 97 + 98 + 99 = 121
()
October 30, 2007
12 / 12
Exercises:
5.1.7 - If A = {1, 2, 3, 4, 5} and B = {w , x, y , z}, how many
Answer: 220 = 1, 048, 576
elements are there in P(A × B).
5.1.3 - For A = {1, 2, 3} and B = {2, 4, 5}
a) |A × B| =?
Answer: 9
b) # of relations from A to B ?
Answer: 29 = 512
Answer: 29 = 512
c) # of relations on A ?
d) # of relations from A to B
Answer: 27 = 128
that contain (1, 2) and (1, 5) ?
e) # of relations from A to B
that contain exactly five ordered pairs ?
Answer: 95 = 126
f) # of relations on A that
contain at least seven elements ? Answer: 97 + 98 + 99 = 121
()
October 30, 2007
12 / 12
Exercises:
5.1.7 - If A = {1, 2, 3, 4, 5} and B = {w , x, y , z}, how many
Answer: 220 = 1, 048, 576
elements are there in P(A × B).
5.1.3 - For A = {1, 2, 3} and B = {2, 4, 5}
a) |A × B| =?
Answer: 9
b) # of relations from A to B ?
Answer: 29 = 512
Answer: 29 = 512
c) # of relations on A ?
d) # of relations from A to B
Answer: 27 = 128
that contain (1, 2) and (1, 5) ?
e) # of relations from A to B
that contain exactly five ordered pairs ?
Answer: 95 = 126
f) # of relations on A that
contain at least seven elements ? Answer: 97 + 98 + 99 = 121
()
October 30, 2007
12 / 12
Exercises:
5.1.7 - If A = {1, 2, 3, 4, 5} and B = {w , x, y , z}, how many
Answer: 220 = 1, 048, 576
elements are there in P(A × B).
5.1.3 - For A = {1, 2, 3} and B = {2, 4, 5}
a) |A × B| =?
Answer: 9
b) # of relations from A to B ?
Answer: 29 = 512
Answer: 29 = 512
c) # of relations on A ?
d) # of relations from A to B
Answer: 27 = 128
that contain (1, 2) and (1, 5) ?
e) # of relations from A to B
that contain exactly five ordered pairs ?
Answer: 95 = 126
f) # of relations on A that
contain at least seven elements ? Answer: 97 + 98 + 99 = 121
()
October 30, 2007
12 / 12
Exercises:
5.1.7 - If A = {1, 2, 3, 4, 5} and B = {w , x, y , z}, how many
Answer: 220 = 1, 048, 576
elements are there in P(A × B).
5.1.3 - For A = {1, 2, 3} and B = {2, 4, 5}
a) |A × B| =?
Answer: 9
b) # of relations from A to B ?
Answer: 29 = 512
Answer: 29 = 512
c) # of relations on A ?
d) # of relations from A to B
Answer: 27 = 128
that contain (1, 2) and (1, 5) ?
e) # of relations from A to B
that contain exactly five ordered pairs ?
Answer: 95 = 126
f) # of relations on A that
contain at least seven elements ? Answer: 97 + 98 + 99 = 121
()
October 30, 2007
12 / 12
Exercises:
5.1.7 - If A = {1, 2, 3, 4, 5} and B = {w , x, y , z}, how many
Answer: 220 = 1, 048, 576
elements are there in P(A × B).
5.1.3 - For A = {1, 2, 3} and B = {2, 4, 5}
a) |A × B| =?
Answer: 9
b) # of relations from A to B ?
Answer: 29 = 512
Answer: 29 = 512
c) # of relations on A ?
d) # of relations from A to B
Answer: 27 = 128
that contain (1, 2) and (1, 5) ?
e) # of relations from A to B
that contain exactly five ordered pairs ?
Answer: 95 = 126
f) # of relations on A that
contain at least seven elements ? Answer: 97 + 98 + 99 = 121
()
October 30, 2007
12 / 12