• Real Analysis
• 1. Sets and Relations
• 1.1. Notation and Set Theory
• 1.2. Relations and Functions
• 1.3. Equivalence Relations and Classes
• 1.4. Natural Numbers, Integers, and Rational Numbers
• 2. Infinity and Induction
• 3. Sequences of Numbers
• 4. Series of Numbers
• 5. Topology
• 6. Limits, Continuity, and Differentiation
• 7. The Integral
• 8. Sequences of Functions
• 9. Historical Tidbits
• Java Tools

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)

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).

• 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)

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.

Interactive Real Analysis, ver. 2.0.1(c)
(c) 1994-2018, Bert G. Wachsmuth
Page last modified: Jan 26, 2023