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:
  • $|a \cup b \cup c| = |a| + |b| + |c| - |a \cap b| - |b \cap c| - |a \cap c| + |a \cap b \cap c|$