The Boolean Logic of Set Intersections

The intersection of two sets contains those elements that are in both sets. We write the intersection of set A and set B as $A\cap B$.

The intersection of many sets contains those elements that are present in all of them.

Two sets with an empty intersection are called disjoint sets.

Now some food for thought: we can think of the intersection as a binary operator. That is $\cap$ takes two sets as input and spits out a new set as output.

An intersection is an operation between two sets. If these sets are $A$ and $B$, then the intersection of the two sets is a set itself, the set containing all elements in both $A$ and $B$.

The intersection between sets $A$ and $B$ is denoted $A \cap B$. Here is an example:

$A = {6, 7, 8}$

$B = {5, 6, 7}$

$\Downarrow$

$A \cap B = {6, 7}$

Note that $6$ and $7$ are the only elements in both $A$ and $B$, so they are the only elements in both $A$ and $B$ by definition.

Now, here are a few other important properties of sets.

i) Intersections are commutative. Thus for sets $A$ and $B$:

$A \cap B = B \cap A$

ii) Intersections are associative. Thus for sets $A$, $B$, and $C$:

$A \cap (B \cap C) = (A \cap B) \cap C$

iii) In some cases two sets $A$ and $B$ will not share any elements. In this case,

$A \cap B = \emptyset$

and the two sets $A$ and $B$ are called disjoint.

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