Set Intersection/Examples

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Examples of Set Intersection

Example: $2$ Arbitrarily Chosen Sets

Let:

\(\ds S\) \(=\) \(\ds \set {a, b, c}\)
\(\ds T\) \(=\) \(\ds \set {c, e, f, b}\)

Then:

$S \cap T = \set {b, c}$


Example: $2$ Arbitrarily Chosen Sets of Complex Numbers: $1$

Let:

\(\ds A\) \(=\) \(\ds \set {3, -i, 4, 2 + i, 5}\)
\(\ds B\) \(=\) \(\ds \set {-i, 0, -1, 2 + i}\)

Then:

$A \cap B = \set {-i, 2 + i}$


Example: $2$ Arbitrarily Chosen Sets of Complex Numbers: $2$

Let:

\(\ds A\) \(=\) \(\ds \set {3, -i, 4, 2 + i, 5}\)
\(\ds C\) \(=\) \(\ds \set {-\sqrt 2 i, \dfrac 1 2, 3}\)

Then:

$A \cap C = \set 3$


Example: $3$ Arbitrarily Chosen Sets

Let:

\(\ds A_1\) \(=\) \(\ds \set {1, 2, 3, 4}\)
\(\ds A_2\) \(=\) \(\ds \set {1, 2, 5}\)
\(\ds A_3\) \(=\) \(\ds \set {2, 4, 6, 8, 12}\)

Then:

$A_1 \cap A_2 \cap A_3 = \set 2$


Example: $3$ Arbitrarily Chosen Sets of Complex Numbers

Let:

\(\ds A\) \(=\) \(\ds \set {3, -i, 4, 2 + i, 5}\)
\(\ds B\) \(=\) \(\ds \set {-i, 0, -1, 2 + i}\)
\(\ds C\) \(=\) \(\ds \set {-\sqrt 2 i, \dfrac 1 2, 3}\)

Then:

$B \cap C = \O$

and so it follows that:

$A \cap \paren {B \cap C} = \O$


Example: $4$ Arbitrarily Chosen Sets of Complex Numbers

Let:

\(\ds A\) \(=\) \(\ds \set {1, i, -i}\)
\(\ds B\) \(=\) \(\ds \set {2, 1, -i}\)
\(\ds C\) \(=\) \(\ds \set {i, -1, 1 + i}\)
\(\ds D\) \(=\) \(\ds \set {0, -i, 1}\)

Then:

$\paren {A \cup C} \cap \paren {B \cup D} = \set {1, -i}$


Example: Blue-Eyed British People

Let:

\(\ds B\) \(=\) \(\ds \set {\text {British people} }\)
\(\ds C\) \(=\) \(\ds \set {\text {Blue-eyed people} }\)

Then:

$B \cap C = \set {\text {Blue-eyed British people} }$


Example: Overlapping Integer Sets

Let:

\(\ds A\) \(=\) \(\ds \set {x \in \Z: 2 \le x}\)
\(\ds B\) \(=\) \(\ds \set {x \in \Z: x \le 5}\)

Then:

$A \cap B = \set {2, 3, 4, 5}$

and so is finite.


Example: $2$ Circles in Complex Plane

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

\(\ds A\) \(=\) \(\ds \set {z \in \C: \cmod {z - 1} < 3}\)
\(\ds B\) \(=\) \(\ds \set {z \in \C: \cmod {z - 2 i} < 2}\)

Then $A \cap B$ can be illustrated graphically as:

Set-Intersection-Circles-in-Complex-Plane.png

where the intersection is depicted in yellow.


Example: $3$ Circles in Complex Plane

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

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

Then $A \cap B \cap C$ can be illustrated graphically as:

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

where the intersection is depicted in yellow.


Example: Arbitrary Integer Sets

Let:

\(\ds A\) \(=\) \(\ds \set {2, 4, 6, \dotsc}\)
\(\ds C\) \(=\) \(\ds \set {1, 2, 3, 4}\)

Then:

$A \cap C = \set {2, 4}$


Example: Intersection with Power Set

Let $S$ be the set defined as:

$S = \set {1, 2, \set {1, 2} }$


Then the power set of $S$ is:

$\powerset S = \set {\O, \set 1, \set 2, \set {\set {1, 2} }, \set {1, 2}, \set {1, \set {1, 2} }, \set {2, \set {1, 2} }, \set {1, 2, \set {1, 2} } }$


and the intersection of $S$ with $\powerset S$ is:

$S \cap \powerset S = \set {\set {1, 2} }$