Operations on Sets

[MathDefinition title="Union of Sets"] The union of two sets A and B, denoted by A ∪ B, is the set of all elements which are in A, or in B, or in both. A ∪ B = {x : x ∈ A or x ∈ B} [/MathDefinition]

[MathDefinition title="Intersection of Sets"] The intersection of two sets A and B, denoted by A ∩ B, is the set of all elements which are common to both A and B. A ∩ B = {x : x ∈ A and x ∈ B} [/MathDefinition]

[MathDefinition title="Difference of Sets"] The difference of two sets A and B, denoted by A − B, is the set of elements which are in A but not in B. A − B = {x : x ∈ A and x ∉ B} [/MathDefinition]

[MathDefinition title="Complement of a Set"] The complement of a set A, denoted by A', is the set of all elements in the universal set U which are not in A. A' = {x : x ∈ U and x ∉ A} [/MathDefinition]

[MathKeyFormula title="Important Formulas"]

1. A ∪ B = B ∪ A (Commutative law)
2. A ∩ B = B ∩ A (Commutative law)
3. A ∪ (B ∪ C) = (A ∪ B) ∪ C (Associative law)
4. A ∩ (B ∩ C) = (A ∩ B) ∩ C (Associative law)
5. A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C) (Distributive law)
6. A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C) (Distributive law)
7. A − B = A ∩ B'
8. (A')' = A [/MathKeyFormula]

[MathNote title="Quick Tips"] Always draw a Venn diagram to visualize set operations. Remember that A ∪ B includes all elements from both sets while A ∩ B includes only common elements. [/MathNote]

[MathExample title="Example 1: Finding Union"] Problem: If A = {1, 2, 3, 4} and B = {3, 4, 5, 6}, find A ∪ B.

Solution: A ∪ B contains all elements from both sets. A ∪ B = {1, 2, 3, 4} ∪ {3, 4, 5, 6} A ∪ B = {1, 2, 3, 4, 5, 6} Note: Elements 3 and 4 appear in both sets but are listed only once in the union. [/MathExample]

[MathExample title="Example 2: Finding Intersection"] Problem: If A = {1, 2, 3, 4, 5} and B = {2, 4, 6, 8}, find A ∩ B.

Solution: A ∩ B contains only elements common to both sets. A ∩ B = {1, 2, 3, 4, 5} ∩ {2, 4, 6, 8} A ∩ B = {2, 4} Only 2 and 4 appear in both sets. [/MathExample]

[MathExample title="Example 3: Finding Difference"] Problem: If A = {a, b, c, d, e} and B = {b, d, f, g}, find A − B and B − A.

Solution: A − B = {x : x ∈ A and x ∉ B} A − B = {a, b, c, d, e} − {b, d, f, g} A − B = {a, c, e}

B − A = {x : x ∈ B and x ∉ A} B − A = {b, d, f, g} − {a, b, c, d, e} B − A = {f, g}

Note: A − B ≠ B − A in general. [/MathExample]

[MathExample title="Example 4: Finding Complement"] Problem: If U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} and A = {2, 4, 6, 8}, find A'.

Solution: A' = U − A = {x : x ∈ U and x ∉ A} A' = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} − {2, 4, 6, 8} A' = {1, 3, 5, 7, 9, 10} A' contains all elements in U that are not in A. [/MathExample]