The algebraic properties of set operations define how sets behave under operations like union, intersection, and complement, providing a framework for manipulating and simplifying set expressions.
Key Properties
These properties are fundamental to set theory and allow for the manipulation and simplification of complex set expressions. They are analogous to algebraic properties found in arithmetic.
1. Commutative Laws
The order in which sets are combined using union or intersection does not affect the result.
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Union: A ∪ B = B ∪ A (The union of A and B is the same as the union of B and A).
- Example: If A = {1, 2} and B = {3, 4}, then A ∪ B = {1, 2, 3, 4} and B ∪ A = {3, 4, 1, 2} which are equal (order doesn't matter in sets).
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Intersection: A ∩ B = B ∩ A (The intersection of A and B is the same as the intersection of B and A).
- Example: If A = {1, 2, 3} and B = {2, 3, 4}, then A ∩ B = {2, 3} and B ∩ A = {2, 3}.
2. Associative Laws
When performing multiple unions or intersections, the grouping of sets does not affect the result.
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Union: (A ∪ B) ∪ C = A ∪ (B ∪ C)
- Example: If A = {1}, B = {2}, and C = {3}, then (A ∪ B) ∪ C = ({1} ∪ {2}) ∪ {3} = {1, 2} ∪ {3} = {1, 2, 3} and A ∪ (B ∪ C) = {1} ∪ ({2} ∪ {3}) = {1} ∪ {2, 3} = {1, 2, 3}.
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Intersection: (A ∩ B) ∩ C = A ∩ (B ∩ C)
- Example: If A = {1, 2}, B = {2, 3}, and C = {2, 4}, then (A ∩ B) ∩ C = ({1, 2} ∩ {2, 3}) ∩ {2, 4} = {2} ∩ {2, 4} = {2} and A ∩ (B ∩ C) = {1, 2} ∩ ({2, 3} ∩ {2, 4}) = {1, 2} ∩ {2} = {2}.
3. Distributive Laws
These laws describe how union distributes over intersection, and vice versa.
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Union over Intersection: A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)
- Example: If A = {1}, B = {1, 2}, and C = {1, 3}, then A ∪ (B ∩ C) = {1} ∪ ({1, 2} ∩ {1, 3}) = {1} ∪ {1} = {1} and (A ∪ B) ∩ (A ∪ C) = ({1} ∪ {1, 2}) ∩ ({1} ∪ {1, 3}) = {1, 2} ∩ {1, 3} = {1}.
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Intersection over Union: A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)
- Example: If A = {1, 2}, B = {1}, and C = {3}, then A ∩ (B ∪ C) = {1, 2} ∩ ({1} ∪ {3}) = {1, 2} ∩ {1, 3} = {1} and (A ∩ B) ∪ (A ∩ C) = ({1, 2} ∩ {1}) ∪ ({1, 2} ∩ {3}) = {1} ∪ {} = {1}.
4. Identity Laws
These laws identify the identity elements for union and intersection.
- Union: A ∪ ∅ = A (The union of a set A with the empty set is A). The empty set is the identity element for the union operation.
- Intersection: A ∩ U = A (The intersection of a set A with the universal set is A). The universal set is the identity element for the intersection operation.
5. Complement Laws
These laws describe the interaction between a set and its complement.
- A ∪ A' = U (The union of a set A with its complement is the universal set).
- A ∩ A' = ∅ (The intersection of a set A with its complement is the empty set).
- (A')' = A (The complement of the complement of A is A).
6. Idempotent Laws
Applying the same set operation multiple times yields the same result as applying it once.
- Union: A ∪ A = A
- Intersection: A ∩ A = A
7. DeMorgan's Laws
These laws provide a way to express the complement of a union or intersection.
- (A ∪ B)' = A' ∩ B' (The complement of the union of A and B is the intersection of the complements of A and B).
- (A ∩ B)' = A' ∪ B' (The complement of the intersection of A and B is the union of the complements of A and B).
8. Domination Laws
- A ∪ U = U (The union of any set with the universal set is the universal set).
- A ∩ ∅ = ∅ (The intersection of any set with the empty set is the empty set).
9. Absorption Laws
These laws describe how union and intersection "absorb" each other under certain conditions.
- A ∪ (A ∩ B) = A
- A ∩ (A ∪ B) = A
These properties are essential for reasoning about sets and performing set operations effectively. They allow us to simplify complex expressions and solve problems involving sets in a systematic way.
