The language of set theory � Sets and elements
� S1 = {a, b, c, d}, S2 = {a, b, e}. � S1 is a set. S2 is a set. � a is an element of S1 .
� ∈: “is in”, ∉: “is not in” � a ∈ S1, a ∈ S2 � e ∉ S1, e ∈ S2
� Empty set � S3 = {}.
The language of set theory � Let S1 = {a, b, c, d, e}, S2 = {a, b, e}.
� ⊂ : “subset” (or ⊆)
� =: “equal sets” � �
� Let S3 = {e, b, d, c, a}. Then S1=S3.
S1
S2
A = B means 8x 2 A, x is also 2 B and; 8y 2 B, y is also 2 A.
In other words, A ⇢ B and B ⇢ A.
S2 ⇢ S1 means
8x 2 S2, x is also 2 S1.
The language of set theory � Let S1 = {a, b, c, d}, S2 = {a, b, e}.
� ∪: “union set” � S1∪ S2 = {a, b, c, d, e}
� ∩: “intersection set” � S1∩ S2 = {a, b}
� \: “difference set” � S1∖ S2 = {c, d} � S2∖ S1 = {e}
S1 S2
S1 ∩ S2
S1 ∪ S2
S1∖ S2 S2∖ S1
Sets that contain infinite elements
4
� R is the set that contain all real numbers.
� R2 is the set the contain all real-valued column vectors whose has two components. �
� Rn is the set the contain all real-valued column vectors whose number of components is n. �
100�9.1
�
32
�
· · ·
1.1
2.2
�
12
�34
�00
�
1�1
�
2
�1�
10
�9�
1⇡
�
R2
1
3
4.2
�5.1
⇡
p2
34
2664
1234
37752
664
�1e⇡0
3
7752664
0000
3775
2
664
4278
3
775
R4
R
· · ·
� Let L denote the line on the xy-plane whose equation is x + y = 1.
� Then L is a subset of R2 and
y 1 -1
Sets that contain infinite elements
5
-1 0 1 x
L R2
� Is R2 a subset of R3 ?
� Is the plane determined by equation “z = 0” a subset of R3 ?
� How do we express the plane “z = 0” as a set?
Questions
6
1 0 1 x
z 1 -1
8<
:
2
4x
y
z
3
5 : x, y, z 2 R, z = 0
9=
;
� What is the difference between the following two sets?
� How about ?
� Is a subset of ? (i.e., is true?)
� Is a subset of ?
Questions
7
L4 =
⇢10
�+ t
1�1
�: t 2 R
�L3 =�v : v 2 R2, Av = 1 where A =
⇥1 1
⇤