Set Operations and Venn Diagrams

Venn Diagrams

Wind diagram is a geometrical method to represent set or sets, using a box as our population or universal set and circular shapes to represent sets inside the box.

• Universal Set: The set of all elements or objects under consideration.
  • Denoted by u in set theory.
  • Referred to as a sample space in probability.

Set Operations

Considering two sets a and b from a universal set u:

• Complement of a Set (a' or a^c):

  • The set of all elements in u but not in a.
  • Set builder notation: \lbrace x | x binom{e}{u} \text{ and } x \notin a \rbrace

• Union of Two Sets (a \cup b):

  • The set of all elements that are in a or b or both.
  • Set builder notation: \lbrace x | x binom{e}{a} \text{ or } x binom{e}{b} \rbrace

• Intersection of Two Sets (a \cap b):

  • The set of all elements that are in both a and b at the same time (overlap area).
  • Set builder notation: \lbrace x | x binom{e}{a} \text{ and } x binom{e}{b} \rbrace

• Difference of Two Sets (a - b):

  • The set of all elements that are in a but not in b.
  • Set builder notation: \lbrace x | x binom{e}{a} \text{ and } x \notin b \rbrace

• Cartesian Product (a \times b):

  • The set of all ordered pairs (x, y) where x belongs to a and y belongs to b.
  • a \times b \neq b \times a

Cardinality

• Cardinality of a Union of Two Sets:
  • |a \cup b| = |a| + |b| - |a \cap b|
• Cardinality of a Union of Three Sets:
  • |a \cup b \cup c| = |a| + |b| + |c| - |a \cap b| - |b \cap c| - |a \cap c| + |a \cap b \cap c|