Next: Union
Up: Operations with sets
Previous: Equality
Contents
Definition 3.2.7
Let
and
be two sets. The intersection of
and
is the
set of all the elements that
and
have in common. We denote:
Note that is a subset of and a subset of .
Example 3.2.8
Let
and
. Then
.
The intersection of two sets can be repesented using Venn diagrams:
Figure 3:
The intersection of two sets

Proposition 3.2.9 (the commutative law)
Let
and
be two sets. Then the following equality holds:
Proof.
We use truth table with the following notations:
Thus,
is equivalent to
and
is equivalent to
. Our proposition is equivalent to the commutative law for the logical connector
(v.s. Proposition prop commutative law for and).
The commutative law for the intersection of sets can be generalized to the intersection of any (finite) number of sets; the proof is done by induction (v.i. Prop.Definition def Peano).
Proposition 3.2.10 (the associative law)
Let
,
and
be three sets. Then the following equality holds:
We prove it using truth tables, Proposition prop associative law for or, with notations similar to those in the proof of Proposition 2.14.
Figure 4:
Intersection: associative law  first part

Figure 5:
Intersection: asociative law  second part

The associative law for the intersection of sets can be generalized to the
intersection of any (finite) number of sets; the proof is done by induction (v.i. Prop. def Peano).
An extension of the associative law for intersection is as follows. Let and be two sets of indices; if
and
are two families of sets, then
Proposition 3.2.11
Let
be any set. Then we have:
This proposition can be proven with truth tables.
Proposition 3.2.12
For any two sets, the following holds:
Here too, use truth tables.
Next: Union
Up: Operations with sets
Previous: Equality
Contents
root
20020610