Set Intersection/Examples
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Examples of Set Intersection
Example: $2$ Arbitrarily Chosen Sets: $1$
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: $2$
Let:
| \(\ds A\) | \(=\) | \(\ds \set {1, 2, 3, 4, 5, 6}\) | ||||||||||||
| \(\ds B\) | \(=\) | \(\ds \set {1, 4, 5, 6, 7, 8}\) |
Then:
- $A \cap B = \set {1, 4, 5, 6}$
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:
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:
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:
- $S \cap \powerset S = \set {\set {1, 2} }$
Arbitrary Example $1$
Let:
| \(\ds A\) | \(=\) | \(\ds \set {1, 2}\) | ||||||||||||
| \(\ds B\) | \(=\) | \(\ds \set {2, 3}\) |
Then:
- $A \cap B = \set 2$