Set Theory/Examples

Examples in Set Theory

Unions and Intersections $1$

Let:

\(\ds V_1\)

\(=\)

\(\ds \set {v_1, v_3, v_4}\)

\(\ds V_2\)

\(=\)

\(\ds \set {v_2, v_5}\)

\(\ds V_3\)

\(=\)

\(\ds \set {v_1, v_3}\)

Then:

\(\ds V_1 \cup V_2\)

\(=\)

\(\ds \set {v_1, v_2, v_3, v_4, v_5}\)

\(\ds V_1 \cup V_3\)

\(=\)

\(\ds \set {v_1, v_3, v_4}\)

\(\ds V_2 \cup V_3\)

\(=\)

\(\ds \set {v_1, v_2, v_3, v_5}\)

\(\ds V_1 \cap V_2\)

\(=\)

\(\ds \O\)

\(\ds V_1 \cap V_3\)

\(=\)

\(\ds \set {v_1, v_3}\)

\(\ds V_2 \cap V_3\)

\(=\)

\(\ds \O\)

Thus:

$V_1$ and $V_2$ are disjoint

$V_2$ and $V_3$ are disjoint.

Unions and Intersections $2$

Let:

\(\ds A\)

\(=\)

\(\ds \set {1, 2}\)

\(\ds B\)

\(=\)

\(\ds \set {1, \set 2}\)

\(\ds C\)

\(=\)

\(\ds \set {\set 1, \set 2}\)

\(\ds D\)

\(=\)

\(\ds \set {\set 1, \set 2, \set {1, 2} }\)

Then:

\(\ds A \cap B\)

\(=\)

\(\ds \set 1\)

\(\ds \paren {B \cap D} \cup A\)

\(=\)

\(\ds \set {1, 2, \set 2}\)

\(\ds \paren {A \cap B} \cup D\)

\(=\)

\(\ds \set {1, \set 1, \set 2, \set {1, 2} }\)

\(\ds \paren {A \cap B} \cup \paren {C \cap D}\)

\(=\)

\(\ds \set {1, \set 1, \set 2}\)

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:

\(\ds A\)

\(=\)

\(\ds \set {z \in \C: \cmod {z + i} < 3}\)

\(\ds B\)

\(=\)

\(\ds \set {z \in \C: \cmod z < 5}\)

\(\ds C\)

\(=\)

\(\ds \set {z \in \C: \cmod {z + 1} < 4}\)

Example of Intersection with Union

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

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:

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:

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:

where the union of the intersections is depicted in yellow.