In set theory, the distributive laws describe how the operations of union and intersection interact with each other. These laws help in simplifying complex set expressions and are analogous to distributive properties in arithmetic.
The distributive laws show that one set operation can be "distributed" across another when applied to a group of sets. Let’s explore both laws:
This means that if you take the union of sets B and C, and then find the intersection with set A, you’ll get the same result as if you had intersected A with B and A with C separately, then taken the union of those two results. This law is helpful in breaking down and simplifying set expressions.
This states that if you take the intersection of B and C first, and then union the result with A, it is the same as unioning A with B and A with C separately, then taking the intersection of those two outcomes.
Suppose we have the following sets:
A = {1, 2, 3}, B = {2, 3, 4} and C = {3, 4, 5}
Now, let's demonstrate how the first distributive law works using an example:
The union of sets B and C gives: B ∪ C = {2, 3, 4, 5}
Intersecting this result with set A yields: A ∩ (B ∪ C) = {2, 3}
A ∩ B = {2, 3}
A ∩ C = {3}
(A ∩ B) ∪ (A ∩ C) = {2, 3}
Both sides give {2, 3}, confirming the distributive law holds true.
Distributive laws are vital in simplifying expressions in set theory, logic, database queries, and digital circuit design. Mastery of these rules is essential for problem solving and proving set identities.
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