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## Intersection

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:

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.

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 2002-06-10