Sec. 2.2 Subsets Liberal Arts Math

1
Sec. 2.2 Subsets Liberal Arts Math
def Set A is a subset of set B, denoted A
B , if and only if all the elements of set A
are also elements of set B
2
Sec. 2.2 Subsets Liberal Arts Math
def Set A is a subset of set B, denoted A
B , if and only if all the elements of set A
are also elements of set B
ex. 1 Determine whether set A is a subset of
set B. a) A b, d, g
B a, b, c, d, e, f, g b) A 1, 2,
3, 4, 5, 6 B 2, 4, 6 c) A
?,?,? B ?, ?, ?
3
Sec. 2.2 Subsets Liberal Arts Math
def Set A is a subset of set B, denoted A
B , if and only if all the elements of set A
are also elements of set B
ex. 1 Determine whether set A is a subset of
set B. a) A b, d, g
B a, b, c, d, e, f, g b) A 1, 2,
3, 4, 5, 6 B 2, 4, 6 c) A
?,?,? B ?, ?, ?
A B
4
Sec. 2.2 Subsets Liberal Arts Math
def Set A is a subset of set B, denoted A
B , if and only if all the elements of set A
are also elements of set B
ex. 1 Determine whether set A is a subset of
set B. a) A b, d, g
B a, b, c, d, e, f, g b) A 1, 2,
3, 4, 5, 6 B 2, 4, 6 c) A
?,?,? B ?, ?, ?
A B
A B
5
Sec. 2.2 Subsets Liberal Arts Math
def Set A is a subset of set B, denoted A
B , if and only if all the elements of set A
are also elements of set B
ex. 1 Determine whether set A is a subset of
set B. a) A b, d, g
B a, b, c, d, e, f, g b) A 1, 2,
3, 4, 5, 6 B 2, 4, 6 c) A
?,?,? B ?, ?, ?
A B
A B
A B
6
Sec. 2.2 Subsets Liberal Arts Math
def Set A is a proper subset of set B,
denoted A B, iff (if and only if) all
the elements of set A are elements of set B
and set A set B
7
Sec. 2.2 Subsets Liberal Arts Math
def Set A is a proper subset of set B,
denoted A B, iff (if and only if) all
the elements of set A are elements of set B
and set A set B (ie., set B
must contain at least one element not in set A).
8
Sec. 2.2 Subsets Liberal Arts Math
def Set A is a proper subset of set B,
denoted A B, iff (if and only if) all
the elements of set A are elements of set B
and set A set B (ie., set B
must contain at least one element not in set A).
ex. 2 Determine whether set A is a proper
subset of set B. a) A b, d, g B a,
b, c, d, e, f, g b) A 1, 2, 3, 4,
5, 6 B 2, 4, 6 c) A
?,?,? B ?, ?, ?
9
Sec. 2.2 Subsets Liberal Arts Math
def Set A is a proper subset of set B,
denoted A B, iff (if and only if) all
the elements of set A are elements of set B
and set A set B (ie., set B
must contain at least one element not in set A).
ex. 2 Determine whether set A is a proper
subset of set B. a) A b, d, g B a,
b, c, d, e, f, g b) A 1, 2, 3, 4,
5, 6 B 2, 4, 6 c) A
?,?,? B ?, ?, ?
A B
10
Sec. 2.2 Subsets Liberal Arts Math
def Set A is a proper subset of set B,
denoted A B, iff (if and only if) all
the elements of set A are elements of set B
and set A set B (ie., set B
must contain at least one element not in set A).
ex. 2 Determine whether set A is a proper
subset of set B. a) A b, d, g B a,
b, c, d, e, f, g b) A 1, 2, 3, 4,
5, 6 B 2, 4, 6 c) A
?,?,? B ?, ?, ?
A B
A B
11
Sec. 2.2 Subsets Liberal Arts Math
def Set A is a proper subset of set B,
denoted A B, iff (if and only if) all
the elements of set A are elements of set B
and set A set B (ie., set B
must contain at least one element not in set A).
ex. 2 Determine whether set A is a proper
subset of set B. a) A b, d, g B a,
b, c, d, e, f, g b) A 1, 2, 3, 4,
5, 6 B 2, 4, 6 c) A
?,?,? B ?, ?, ?
A B
A B
A B
12
Sec. 2.2 Subsets Liberal Arts Math

• ex. 3 Determine whether the following are True
  or False
• 3 1, 2, 3
• 3 1, 2, 3 e) 3 1, 2, 3
• 3 f) 3 1, 2, 3
• d) 3 1, 2, 3 g) 1, 2, 3

13
Sec. 2.2 Subsets Liberal Arts Math

• ex. 3 Determine whether the following are True
  or False
• 3 1, 2, 3
• 3 1, 2, 3 e) 3 1, 2, 3
• 3 f) 3 1, 2, 3
• d) 3 1, 2, 3 g) 1, 2, 3

True
14
Sec. 2.2 Subsets Liberal Arts Math

• ex. 3 Determine whether the following are True
  or False
• 3 1, 2, 3
• 3 1, 2, 3 e) 3 1, 2, 3
• 3 f) 3 1, 2, 3
• d) 3 1, 2, 3 g) 1, 2, 3

True
False
15
Sec. 2.2 Subsets Liberal Arts Math

• ex. 3 Determine whether the following are True
  or False
• 3 1, 2, 3
• 3 1, 2, 3 e) 3 1, 2, 3
• 3 f) 3 1, 2, 3
• d) 3 1, 2, 3 g) 1, 2, 3

True
False
True
16
Sec. 2.2 Subsets Liberal Arts Math

• ex. 3 Determine whether the following are True
  or False
• 3 1, 2, 3
• 3 1, 2, 3 e) 3 1, 2, 3
• 3 f) 3 1, 2, 3
• d) 3 1, 2, 3 g) 1, 2, 3

True
False
True
False
17
Sec. 2.2 Subsets Liberal Arts Math

• ex. 3 Determine whether the following are True
  or False
• 3 1, 2, 3
• 3 1, 2, 3 e) 3 1, 2, 3
• 3 f) 3 1, 2, 3
• d) 3 1, 2, 3 g) 1, 2, 3

True
False
True
True
False
18
Sec. 2.2 Subsets Liberal Arts Math

• ex. 3 Determine whether the following are True
  or False
• 3 1, 2, 3
• 3 1, 2, 3 e) 3 1, 2, 3
• 3 f) 3 1, 2, 3
• d) 3 1, 2, 3 g) 1, 2, 3

True
False
True
True
True
False
19
Sec. 2.2 Subsets Liberal Arts Math

• ex. 3 Determine whether the following are True
  or False
• 3 1, 2, 3
• 3 1, 2, 3 e) 3 1, 2, 3
• 3 f) 3 1, 2, 3
• d) 3 1, 2, 3 g) 1, 2, 3

True
False
True
True
True
False
True
20
Sec. 2.2 Subsets Liberal Arts Math
Number
of set distinct subsets Note
1 2

21
Sec. 2.2 Subsets Liberal Arts Math
Number
of set distinct subsets Note
1 2 , 1

22
Sec. 2.2 Subsets Liberal Arts Math
Number
of set distinct subsets Note
1 2 , 1 1, 2
4 1, 2, 3 1, 2, 3, 4 1,
2, 3, 4, 5
23
Sec. 2.2 Subsets Liberal Arts Math
Number
of set distinct subsets Note
1 2 , 1 1, 2
4 1, 2, 3 1,
2, 3, 4 1, 2, 3, 4, 5

, 1, 2, 1, 2
24
Sec. 2.2 Subsets Liberal Arts Math
Number
of set distinct subsets Note
1 2 , 1 1, 2
4 1, 2, 3 8 1,
2, 3, 4 1, 2, 3, 4, 5

, 1, 2, 1, 2
25
Sec. 2.2 Subsets Liberal Arts Math
Number
of set distinct subsets Note
1 2 , 1 1, 2
4 1, 2, 3 8 1,
2, 3, 4 1, 2, 3, 4, 5

, 1, 2, 1, 2

26
Sec. 2.2 Subsets Liberal Arts Math
Number
of set distinct subsets Note
1 2 , 1 1, 2
4 1, 2, 3 8 1,
2, 3, 4 1, 2, 3, 4, 5

, 1, 2, 1, 2
, 1
27
Sec. 2.2 Subsets Liberal Arts Math
Number
of set distinct subsets Note
1 2 , 1 1, 2
4 1, 2, 3 8 1,
2, 3, 4 1, 2, 3, 4, 5

, 1, 2, 1, 2
, 1, 2
28
Sec. 2.2 Subsets Liberal Arts Math
Number
of set distinct subsets Note
1 2 , 1 1, 2
4 1, 2, 3 8 1,
2, 3, 4 1, 2, 3, 4, 5

, 1, 2, 1, 2
, 1, 2, 3,
29
Sec. 2.2 Subsets Liberal Arts Math
Number
of set distinct subsets Note
1 2 , 1 1, 2
4 1, 2, 3 8 1,
2, 3, 4 1, 2, 3, 4, 5

, 1, 2, 1, 2
, 1, 2, 3, 1, 2
30
Sec. 2.2 Subsets Liberal Arts Math
Number
of set distinct subsets Note
1 2 , 1 1, 2
4 1, 2, 3 8 1,
2, 3, 4 1, 2, 3, 4, 5

, 1, 2, 1, 2
, 1, 2, 3, 1, 2, 1, 3
31
Sec. 2.2 Subsets Liberal Arts Math
Number
of set distinct subsets Note
1 2 , 1 1, 2
4 1, 2, 3 8 1,
2, 3, 4 1, 2, 3, 4, 5

, 1, 2, 1, 2
, 1, 2, 3, 1, 2, 1, 3, 2, 3
32
Sec. 2.2 Subsets Liberal Arts Math
Number
of set distinct subsets Note
1 2 , 1 1, 2
4 1, 2, 3 8 1,
2, 3, 4 1, 2, 3, 4, 5

, 1, 2, 1, 2
, 1, 2, 3, 1, 2, 1, 3, 2, 3,
1, 2, 3
33
Sec. 2.2 Subsets Liberal Arts Math
Number
of set distinct subsets Note
1 2 , 1 1, 2
4 1, 2, 3 8 1,
2, 3, 4 16 1, 2, 3, 4, 5

, 1, 2, 1, 2
, 1, 2, 3, 1, 2, 1, 3, 2, 3,
1, 2, 3
34
Sec. 2.2 Subsets Liberal Arts Math
Number
of set distinct subsets Note
1 2 , 1 1, 2
4 1, 2, 3 8 1,
2, 3, 4 16 1, 2, 3, 4, 5

, 1, 2, 1, 2
, 1, 2, 3, 1, 2, 1, 3, 2, 3,
1, 2, 3

35
Sec. 2.2 Subsets Liberal Arts Math
Number
of set distinct subsets Note
1 2 , 1 1, 2
4 1, 2, 3 8 1,
2, 3, 4 16 1, 2, 3, 4, 5

, 1, 2, 1, 2
, 1, 2, 3, 1, 2, 1, 3, 2, 3,
1, 2, 3
, 1
36
Sec. 2.2 Subsets Liberal Arts Math
Number
of set distinct subsets Note
1 2 , 1 1, 2
4 1, 2, 3 8 1,
2, 3, 4 16 1, 2, 3, 4, 5

, 1, 2, 1, 2
, 1, 2, 3, 1, 2, 1, 3, 2, 3,
1, 2, 3
, 1, 2
37
Sec. 2.2 Subsets Liberal Arts Math
Number
of set distinct subsets Note
1 2 , 1 1, 2
4 1, 2, 3 8 1,
2, 3, 4 16 1, 2, 3, 4, 5

, 1, 2, 1, 2
, 1, 2, 3, 1, 2, 1, 3, 2, 3,
1, 2, 3
, 1, 2, 3
38
Sec. 2.2 Subsets Liberal Arts Math
Number
of set distinct subsets Note
1 2 , 1 1, 2
4 1, 2, 3 8 1,
2, 3, 4 16 1, 2, 3, 4, 5

, 1, 2, 1, 2
, 1, 2, 3, 1, 2, 1, 3, 2, 3,
1, 2, 3
, 1, 2, 3, 4
39
Sec. 2.2 Subsets Liberal Arts Math
Number
of set distinct subsets Note
1 2 , 1 1, 2
4 1, 2, 3 8 1,
2, 3, 4 16 1, 2, 3, 4, 5

, 1, 2, 1, 2
, 1, 2, 3, 1, 2, 1, 3, 2, 3,
1, 2, 3
, 1, 2, 3, 4, 1, 2
40
Sec. 2.2 Subsets Liberal Arts Math
Number
of set distinct subsets Note
1 2 , 1 1, 2
4 1, 2, 3 8 1,
2, 3, 4 16 1, 2, 3, 4, 5

, 1, 2, 1, 2
, 1, 2, 3, 1, 2, 1, 3, 2, 3,
1, 2, 3
, 1, 2, 3, 4, 1, 2, 1, 3
41
Sec. 2.2 Subsets Liberal Arts Math
Number
of set distinct subsets Note
1 2 , 1 1, 2
4 1, 2, 3 8 1,
2, 3, 4 16 1, 2, 3, 4, 5

, 1, 2, 1, 2
, 1, 2, 3, 1, 2, 1, 3, 2, 3,
1, 2, 3
, 1, 2, 3, 4, 1, 2, 1, 3, 1, 4

42
Sec. 2.2 Subsets Liberal Arts Math
Number
of set distinct subsets Note
1 2 , 1 1, 2
4 1, 2, 3 8 1,
2, 3, 4 16 1, 2, 3, 4, 5

, 1, 2, 1, 2
, 1, 2, 3, 1, 2, 1, 3, 2, 3,
1, 2, 3
, 1, 2, 3, 4, 1, 2, 1, 3, 1, 4,
2, 3
43
Sec. 2.2 Subsets Liberal Arts Math
Number
of set distinct subsets Note
1 2 , 1 1, 2
4 1, 2, 3 8 1,
2, 3, 4 16 1, 2, 3, 4, 5

, 1, 2, 1, 2
, 1, 2, 3, 1, 2, 1, 3, 2, 3,
1, 2, 3
, 1, 2, 3, 4, 1, 2, 1, 3, 1, 4,
2, 3, 2, 4
44
Sec. 2.2 Subsets Liberal Arts Math
Number
of set distinct subsets Note
1 2 , 1 1, 2
4 1, 2, 3 8 1,
2, 3, 4 16 1, 2, 3, 4, 5

, 1, 2, 1, 2
, 1, 2, 3, 1, 2, 1, 3, 2, 3,
1, 2, 3
, 1, 2, 3, 4, 1, 2, 1, 3, 1, 4,
2, 3, 2, 4, 3, 4
45
Sec. 2.2 Subsets Liberal Arts Math
Number
of set distinct subsets Note
1 2 , 1 1, 2
4 1, 2, 3 8 1,
2, 3, 4 16 1, 2, 3, 4, 5

, 1, 2, 1, 2
, 1, 2, 3, 1, 2, 1, 3, 2, 3,
1, 2, 3
, 1, 2, 3, 4, 1, 2, 1, 3, 1, 4,
2, 3, 2, 4, 3, 4, 1, 2, 3
46
Sec. 2.2 Subsets Liberal Arts Math
Number
of set distinct subsets Note
1 2 , 1 1, 2
4 1, 2, 3 8 1,
2, 3, 4 16 1, 2, 3, 4, 5

, 1, 2, 1, 2
, 1, 2, 3, 1, 2, 1, 3, 2, 3,
1, 2, 3
, 1, 2, 3, 4, 1, 2, 1, 3, 1, 4,
2, 3, 2, 4, 3, 4, 1, 2, 3, 1, 2, 4

47
Sec. 2.2 Subsets Liberal Arts Math
Number
of set distinct subsets Note
1 2 , 1 1, 2
4 1, 2, 3 8 1,
2, 3, 4 16 1, 2, 3, 4, 5

, 1, 2, 1, 2
, 1, 2, 3, 1, 2, 1, 3, 2, 3,
1, 2, 3
, 1, 2, 3, 4, 1, 2, 1, 3, 1, 4,
2, 3, 2, 4, 3, 4, 1, 2, 3, 1, 2, 4 ,
1,3,4
48
Sec. 2.2 Subsets Liberal Arts Math
Number
of set distinct subsets Note
1 2 , 1 1, 2
4 1, 2, 3 8 1,
2, 3, 4 16 1, 2, 3, 4, 5

, 1, 2, 1, 2
, 1, 2, 3, 1, 2, 1, 3, 2, 3,
1, 2, 3
, 1, 2, 3, 4, 1, 2, 1, 3, 1, 4,
2, 3, 2, 4, 3, 4, 1, 2, 3, 1, 2, 4 ,
1,3,4, 2, 3, 4
49
Sec. 2.2 Subsets Liberal Arts Math
Number
of set distinct subsets Note
1 2 , 1 1, 2
4 1, 2, 3 8 1,
2, 3, 4 16 1, 2, 3, 4, 5

, 1, 2, 1, 2
, 1, 2, 3, 1, 2, 1, 3, 2, 3,
1, 2, 3
, 1, 2, 3, 4, 1, 2, 1, 3, 1, 4,
2, 3, 2, 4, 3, 4, 1, 2, 3, 1, 2, 4 ,
1,3,4, 2, 3, 4, 1, 2, 3, 4
50
Sec. 2.2 Subsets Liberal Arts Math
Number
of set distinct subsets Note
1 2 , 1 1, 2
4 1, 2, 3 8 1,
2, 3, 4 16 1, 2, 3, 4, 5
32
, 1, 2, 1, 2
, 1, 2, 3, 1, 2, 1, 3, 2, 3,
1, 2, 3
, 1, 2, 3, 4, 1, 2, 1, 3, 1, 4,
2, 3, 2, 4, 3, 4, 1, 2, 3, 1, 2, 4 ,
1,3,4, 2, 3, 4, 1, 2, 3, 4
51
Sec. 2.2 Subsets Liberal Arts Math
Number
of set distinct subsets Note
1 2 , 1 1, 2
4 1, 2, 3 8 1,
2, 3, 4 16 1, 2, 3, 4, 5
32
, 1, 2, 1, 2
, 1, 2, 3, 1, 2, 1, 3, 2, 3,
1, 2, 3
, 1, 2, 3, 4, 1, 2, 1, 3, 1, 4,
2, 3, 2, 4, 3, 4, 1, 2, 3, 1, 2, 4 ,
1,3,4, 2, 3, 4, 1, 2, 3, 4
1, 2, 3, 4, 5,, 10
52
Sec. 2.2 Subsets Liberal Arts Math
Number
of set distinct subsets Note
1 2 , 1 1, 2
4 1, 2, 3 8 1,
2, 3, 4 16 1, 2, 3, 4, 5
32
, 1, 2, 1, 2
, 1, 2, 3, 1, 2, 1, 3, 2, 3,
1, 2, 3
, 1, 2, 3, 4, 1, 2, 1, 3, 1, 4,
2, 3, 2, 4, 3, 4, 1, 2, 3, 1, 2, 4 ,
1,3,4, 2, 3, 4, 1, 2, 3, 4
1, 2, 3, 4, 5,, 10
53
Sec. 2.2 Subsets Liberal Arts Math
Number
of set distinct subsets Note
1 2 , 1 1, 2
4 1, 2, 3 8 1,
2, 3, 4 16 1, 2, 3, 4, 5
32
, 1, 2, 1, 2
2x2
, 1, 2, 3, 1, 2, 1, 3, 2, 3,
1, 2, 3
, 1, 2, 3, 4, 1, 2, 1, 3, 1, 4,
2, 3, 2, 4, 3, 4, 1, 2, 3, 1, 2, 4 ,
1,3,4, 2, 3, 4, 1, 2, 3, 4
1, 2, 3, 4, 5,, 10
54
Sec. 2.2 Subsets Liberal Arts Math
Number
of set distinct subsets Note
1 2 , 1 1, 2
4 1, 2, 3 8 1,
2, 3, 4 16 1, 2, 3, 4, 5
32
, 1, 2, 1, 2
2x2
, 1, 2, 3, 1, 2, 1, 3, 2, 3,
1, 2, 3
2x2 x 2
, 1, 2, 3, 4, 1, 2, 1, 3, 1, 4,
2, 3, 2, 4, 3, 4, 1, 2, 3, 1, 2, 4 ,
1,3,4, 2, 3, 4, 1, 2, 3, 4
1, 2, 3, 4, 5,, 10
55
Sec. 2.2 Subsets Liberal Arts Math
Number
of set distinct subsets Note
1 2 , 1 1, 2
4 1, 2, 3 8 1,
2, 3, 4 16 1, 2, 3, 4, 5
32
, 1, 2, 1, 2
2x2
, 1, 2, 3, 1, 2, 1, 3, 2, 3,
1, 2, 3
2x2 x 2
, 1, 2, 3, 4, 1, 2, 1, 3, 1, 4,
2, 3, 2, 4, 3, 4, 1, 2, 3, 1, 2, 4 ,
1,3,4, 2, 3, 4, 1, 2, 3, 4
2x2x2 x 2
1, 2, 3, 4, 5,, 10
56
Sec. 2.2 Subsets Liberal Arts Math
Number
of set distinct subsets Note
1 2 , 1 1, 2
4 1, 2, 3 8 1,
2, 3, 4 16 1, 2, 3, 4, 5
32
, 1, 2, 1, 2
2x2
, 1, 2, 3, 1, 2, 1, 3, 2, 3,
1, 2, 3
2x2 x 2
, 1, 2, 3, 4, 1, 2, 1, 3, 1, 4,
2, 3, 2, 4, 3, 4, 1, 2, 3, 1, 2, 4 ,
1,3,4, 2, 3, 4, 1, 2, 3, 4
2x2x2 x 2
1, 2, 3, 4, 5,, 10
1024
210