1.1. Notation and Set Theory

Proposition 1.1.3: Distributive Law for Sets

A (B C) = (A B) (A C)

A (B C) = (A B) (A C)

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:

To finish the proof, we have to prove the reverse inequality. So, take x in (A union B) intersect (A union C). Then x is in (A or B) as well as in (A or C). Both inequalities together prove equality of the two sets.

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

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