2.1 Union, intersection, and complement operations

Set operations are the building blocks of set theory. They allow us to combine, compare, and manipulate sets in various ways. Understanding these operations is crucial for working with sets and solving complex problems in mathematics and logic.

Union, intersection, and complement are fundamental set operations. The union combines elements from two sets, intersection finds common elements, and complement identifies elements not in a set. These operations form the basis for more advanced set theory concepts and applications.

Set Operations

Combining and Comparing Sets

Top images from around the web for Combining and Comparing Sets

• 

  Set operations illustrated with Venn diagrams | TikZ example View original

  Is this image relevant?

• 

  probability - The union, intersection and complement of events - Cross Validated View original

  Is this image relevant?

• 

  Union, Intersection, and Complement | Mathematics for the Liberal Arts View original

  Is this image relevant?

• 

  Set operations illustrated with Venn diagrams | TikZ example View original

  Is this image relevant?

• 

  probability - The union, intersection and complement of events - Cross Validated View original

  Is this image relevant?

1 of 3

Top images from around the web for Combining and Comparing Sets

• 

  Set operations illustrated with Venn diagrams | TikZ example View original

  Is this image relevant?

• 

  probability - The union, intersection and complement of events - Cross Validated View original

  Is this image relevant?

• 

  Union, Intersection, and Complement | Mathematics for the Liberal Arts View original

  Is this image relevant?

• 

  Set operations illustrated with Venn diagrams | TikZ example View original

  Is this image relevant?

• 

  probability - The union, intersection and complement of events - Cross Validated View original

  Is this image relevant?

1 of 3

• Union of sets A and B, denoted as $A \cup B$, includes all elements that are in either set A or set B or both
• Intersection of sets A and B, denoted as $A \cap B$, includes only the elements that are common to both set A and set B
• Complement of set A, denoted as $A^c$ or $[A'](https://www.fiveableKeyTerm:a')$, includes all elements in the universal set that are not in set A
• Set difference of A and B, denoted as $A - B$ or $A \setminus B$, includes elements in set A that are not in set B
• Symmetric difference of sets A and B, denoted as $A \Delta B$, includes elements that are in either set A or set B but not in both sets
  • Can be expressed as $(A \cup B) - (A \cap B)$ or $(A - B) \cup (B - A)$
  • Example: If $A = \{1, 2, 3\}$ and $B = \{2, 3, 4\}$, then $A \Delta B = \{1, 4\}$

Properties of Set Operations

• Commutative property holds for union and intersection: $A \cup B = B \cup A$ and $A \cap B = B \cap A$
• Associative property holds for union and intersection: $(A \cup B) \cup C = A \cup (B \cup C)$ and $(A \cap B) \cap C = A \cap (B \cap C)$
• Distributive property: $A \cup (B \cap C) = (A \cup B) \cap (A \cup C)$ and $A \cap (B \cup C) = (A \cap B) \cup (A \cap C)$
• De Morgan's laws: $(A \cup B)^c = A^c \cap B^c$ and $(A \cap B)^c = A^c \cup B^c$
  • Example: If $A = \{1, 2, 3\}$ and $B = \{2, 3, 4\}$, then $(A \cup B)^c = \{5, 6, 7, ...\}$ and $A^c \cap B^c = \{5, 6, 7, ...\}$

Set Relationships

Subsets and Proper Subsets

• Set A is a subset of set B, denoted as $A \subseteq B$, if every element of A is also an element of B
  • Example: If $A = \{1, 2\}$ and $B = \{1, 2, 3\}$, then $A \subseteq B$
• Set A is a proper subset of set B, denoted as $A \subset B$, if A is a subset of B and A is not equal to B
  • Implies that set B has at least one element that is not in set A
  • Example: If $A = \{1, 2\}$ and $B = \{1, 2, 3\}$, then $A \subset B$
• Every set is a subset of itself, but not a proper subset of itself
• The empty set is a subset of every set, including itself

Comparing Sets Using Venn Diagrams

• Venn diagrams visually represent relationships between sets using overlapping circles or other shapes
• Subset relationship is represented by one shape completely inside another
• Disjoint sets have no overlapping area in the Venn diagram, indicating no common elements
  • Example: If $A = \{1, 2\}$ and $B = \{3, 4\}$, then A and B are disjoint sets

Special Sets

Universal Set and Its Properties

• The universal set, denoted as U or $\Omega$, is the set of all elements under consideration in a given context
• All other sets in the context are subsets of the universal set
• The complement of the universal set is the empty set: $U^c = \varnothing$
• The union of a set A with its complement results in the universal set: $A \cup A^c = U$
• The intersection of a set A with its complement results in the empty set: $A \cap A^c = \varnothing$

Empty Set and Its Properties

• The empty set, denoted as $\{\}$ or $\varnothing$, is the set containing no elements
• The empty set is a subset of every set, including itself: $\varnothing \subseteq A$ for any set A
• The union of any set A with the empty set results in set A: $A \cup \varnothing = A$
• The intersection of any set A with the empty set results in the empty set: $A \cap \varnothing = \varnothing$
• The complement of the empty set is the universal set: $\varnothing^c = U$