"Unions, intersections, and complements involving 2 sets"

Basic Set Operations

Key Definitions

• Set: A collection of distinct elements.
• Universal Set (U): The set that contains all possible elements in a particular context.
• Subset: A set whose elements are all contained in another set.
• Complement: The complement of a set A, denoted as A', is the set of elements in the universal set that are not in A.

Operations on Sets

1. Intersection (A ∩ B):

   • Definition: The set of elements that are common to both sets A and B.
   • Example: If A = {q, r, y, z} and B = {f, y, z}, then:
     A \cap B = {y,z}

2. Union (A ∪ B):

   • Definition: The set of elements that are in either set A, set B, or both.
   • Example: Using the same sets:
     A \cup B = {f, q, r, y, z}

3. Complement (A'):

   • Definition: Elements in the universal set U that are not in set A.
   • Example: If U = {f, q, r, x, y, z} and A = {q, r, y, z}, then:
     A' = U - A = {f, x}

Finding Complement, Intersection, and Union

• Given:
  • U = {f, q, r, x, y, z}
  • A = {q, r, y, z}
  • B = {f, y, z}

Steps:

1. Finding Complement of A (A'):

   • Using the universal set U,
   • A' = U - A = {f, x}

2. Finding Intersection of A and B (A ∩ B):

   • A \cap B = {y,z}

3. Finding Intersection of A and the Complement of B (A ∩ B'):

   • Knowing that B' is the set of elements not in B:
   • If B = {f, y, z}, then B' = {q, r, x}
   • Therefore,
   • A \cap B' = {q,r,y,z} \cap {q,r,x} = {q,r}

4. Finding Union of A' and B (A' ∪ B):

   • A' \cup B = {f, x} \cup {f, y, z} = {f, x, y, z}

Summary of Results

• A' = {f, x}
• A ∩ B = {y, z}
• A ∩ B' = {q, r}
• A' ∪ B = {f, x, y, z}