# Set Intersection/Examples

## 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: 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:

$A \cap B = \set 2$