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

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32

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· · ·

1.1

2.2

�

12

�34

�00

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1�1

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2

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10

�9�

1⇡

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R2

1

3

4.2

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⇡

p2

34

2664

1234

37752

664

�1e⇡0

3

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0000

3775

2

664

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3

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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

⇤