## 1.1. Notation and Set Theory

### Proposition 1.1.3: Distributive Law for Sets

### Proof:

These relations could be best illustrated by means of a Venn Diagram.**Venn Diagram illustrating**

**A**(**B****C**)

**Venn Diagram for (**

**A****B**) (**A****C**)Obviously, the two resulting sets are the same, hence ‘proving' the first law. However, this is not a rigorous proof, and is therefore not acceptable. Here is a ‘real' proof of the first distribution law:

If *x* is in **A** union (**B** intersect **C**) then *x*
is either in **A** or in (**B** and **C**). Therefore, we have to
consider two cases:

- If
*x*is in**A**, then*x*is also in (**A**union**B**) as well as in (**A**union**C**). Therefore,*x*is in (**A**union**B**) intersect (**A**union**C**). - If
*x*is in (**B**and**C**), then*x*is in (**A**union**B**) because*x*is in**B**, and*x*is also in (**A**union**C**), because*x*is in**C**. Hence, again*x*is in (**A**union**B**) intersect (**A**union**C**). This proves that**A**(**B****C**) (**A****B**) (**A****C**)

*x*in (

**A**union

**B**) intersect (

**A**union

**C**). Then

*x*is in (

**A**or

**B**) as well as in (

**A**or

**C**).

- If
*x*is in**A**, then*x*is also in**A**union (**B**intersect**C**). - If
*x*is in**B**, then it must also be in**C**. Hence,*x*is in**B**intersect**C**, and therefore it is in**A**union (**B**intersect**C**). That shows that**A**(**B****C**) (**A****B**) (**A****C**)

The second distributive laws can be proved the same way, and is left as an exercise.