Set Theory/Examples

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Examples in Set Theory

Unions and Intersections $1$

Let:

\(\displaystyle V_1\) \(=\) \(\displaystyle \set {v_1, v_3, v_4}\)
\(\displaystyle V_2\) \(=\) \(\displaystyle \set {v_2, v_5}\)
\(\displaystyle V_3\) \(=\) \(\displaystyle \set {v_1, v_3}\)


Then:

\(\displaystyle V_1 \cup V_2\) \(=\) \(\displaystyle \set {v_1, v_2, v_3, v_4, v_5}\)
\(\displaystyle V_1 \cup V_3\) \(=\) \(\displaystyle \set {v_1, v_3, v_4}\)
\(\displaystyle V_2 \cup V_3\) \(=\) \(\displaystyle \set {v_1, v_2, v_3, v_5}\)
\(\displaystyle V_1 \cap V_2\) \(=\) \(\displaystyle \O\)
\(\displaystyle V_1 \cap V_3\) \(=\) \(\displaystyle \set {v_1, v_3}\)
\(\displaystyle V_2 \cap V_3\) \(=\) \(\displaystyle \O\)


Thus:

$V_1$ and $V_2$ are disjoint
$V_2$ and $V_3$ are disjoint.


Unions and Intersections $2$

Let:

\(\displaystyle A\) \(=\) \(\displaystyle \set {1, 2}\)
\(\displaystyle B\) \(=\) \(\displaystyle \set {1, \set 2}\)
\(\displaystyle C\) \(=\) \(\displaystyle \set {\set 1, \set 2}\)
\(\displaystyle D\) \(=\) \(\displaystyle \set {\set 1, \set 2, \set {1, 2} }\)


Then:

\(\displaystyle A \cap B\) \(=\) \(\displaystyle \set 1\)
\(\displaystyle \paren {B \cap D} \cup A\) \(=\) \(\displaystyle \set {1, 2, \set 2}\)
\(\displaystyle \paren {A \cap B} \cup D\) \(=\) \(\displaystyle \set {1, \set 1, \set 2, \set {1, 2} }\)
\(\displaystyle \paren {A \cap B} \cup \paren {C \cap D}\) \(=\) \(\displaystyle \set {1, \set 1, \set 2}\)


Equations $A \cup \paren {X \cap B} = C$, $\paren {A \cup X} \cap B = D$

Let $A, B, C, D$ be subsets of a set $S$.

Let there exist $X \subseteq S$ such that:

$A \cup \paren {X \cap B} = C$
$\paren {A \cup X} \cap B = D$

Then:

$A \cap B \subseteq D \subseteq B$

and:

$A \cup D = C$

$\blacksquare$


Simplify $\paren {A \cap B} \cup \paren {C \cap A} \cup \relcomp {\Bbb U} {\relcomp {\Bbb U} A \cup \relcomp {\Bbb U} B}$

The expression:

$\paren {A \cap B} \cup \paren {C \cap A} \cup \relcomp {\Bbb U} {\relcomp {\Bbb U} A \cup \relcomp {\Bbb U} B}$

can be simplified to:

$A \cap \paren {B \cup C}$

$\blacksquare$



Let $A$, $B$ and $C$ be sets defined by circles embedded in the complex plane as follows:

\(\displaystyle A\) \(=\) \(\displaystyle \set {z \in \C: \cmod {z + i} < 3}\)
\(\displaystyle B\) \(=\) \(\displaystyle \set {z \in \C: \cmod z < 5}\)
\(\displaystyle C\) \(=\) \(\displaystyle \set {z \in \C: \cmod {z + 1} < 4}\)


Example of Intersection with Union

$C \cap \paren {A \cup B}$ can be illustrated graphically as:

Set-Intersection-with-Union-3-Circles-in-Complex-Plane.png

where the intersection with the union is depicted in yellow.


Example of Intersection of Unions

$\paren {A \cup B} \cap \paren {B \cup C}$ can be illustrated graphically as:

Set-Intersection-of-Unions-3-Circles-in-Complex-Plane.png

where the intersection of the unions is depicted in yellow.


Example of Union of Intersections

$\paren {A \cap B} \cup \paren {B \cap C} \cup \paren {C \cap A}$ can be illustrated graphically as:

Set-Union-of-Intersections-3-Circles-in-Complex-Plane.png

where the union of the intersections is depicted in yellow.


Example of Union of Intersections with Set Complements

$\paren {A \cap \tilde B} \cup \paren {B \cap \tilde C} \cup \paren {C \cap \tilde A}$, where $\tilde A$ denotes the complement of $A$, can be illustrated graphically as:

Set-Union-of-Intersection-Complements-3-Circles-in-Complex-Plane.png

where the union of the intersections is depicted in yellow.