Intersection

Definition

The intersection of two sets X,Y\mathbb{X}, \mathbb{Y} is the set of all elements that belong to X\mathbb{X} and Y\mathbb{Y}

Formally:

X∩Y={x:x∈X and x∈Y}\mathbb{X} \cap \mathbb{Y} = \{ x : x \in \mathbb{X} \text{ and } x \in \mathbb{Y} \}

This notion of "and" is foundational to computer programming and logic gates.

Properties

For arbitrary sets: X,Y,Z\mathbb{X}, \mathbb{Y}, \mathbb{Z}, and the empty set ∅\emptyset:

i) X∩X=X\mathbb{X} \cap \mathbb{X} = \mathbb{X}

ii) X∩Y=Y∩X\mathbb{X} \cap \mathbb{Y} = \mathbb{Y} \cap \mathbb{X}

iii) X∩(Y∩Z)=(X∩Y)∩Z\mathbb{X} \cap (\mathbb{Y} \cap \mathbb{Z}) = (\mathbb{X} \cap \mathbb{Y}) \cap \mathbb{Z}

iv) X∩∅=∅\mathbb{X} \cap \emptyset = \emptyset

Example

Let X={a,b,c,d}\mathbb{X}={a, b, c, d}, Y={a,e,f}\mathbb{Y}={a, e, f}

Then:

X∩Y={a}\mathbb{X} \cap \mathbb{Y} = \{a\}