Sets and Cardinal Numbers Quiz

Sets and Their Properties

• A universal set ( S ) is represented by a rectangular area, while a subset ( P ) is depicted as a circular shaded region within it. The complement of ( P ) is the unshaded area in ( S ).
• Two sets ( P ) and ( Q ) can be represented with intersecting circles. The union ( P \cup Q ) includes elements from both sets, while the intersection ( P \cap Q ) contains shared elements.

Cardinal Numbers

• The cardinal number ( n(A) ) of a finite set ( A ) indicates the count of distinct elements within the set.
• For example, given ( R = {2, 3, 5, 7} ), the cardinal number is ( n(R) = 4 ).
• The formula for the cardinality of the union of two sets is ( n(A \cup B) = n(A) + n(B) - n(A \cap B) ). If ( A ) and ( B ) are disjoint, ( n(A \cup B) = n(A) + n(B) ).
• For three sets ( P, Q, R ), the cardinality formula is ( n(P \cup Q \cup R) = n(P) + n(Q) + n(R) - n(P \cap Q) - n(Q \cap R) - n(P \cap R) + n(P \cap Q \cap R) ).

Equivalent and Equal Sets

• Two finite sets ( A ) and ( B ) are equivalent if ( n(A) = n(B) ), although they may not be equal. For instance, ( A = {1, 3, 5} ) and ( B = {2, 4, 6} ) are equivalent but not equal.
• Equal sets are when all elements of one set are in another; symbolically ( A = B ) indicates equality.

Power Sets

• The power set ( P(A) ) is the collection of all possible subsets of a set ( A ).
• For example, if ( A = {1, 2, 3} ), then:
  • ( P(A) = { {1, 2, 3}, {1, 2}, {1, 3}, {2, 3}, {1}, {2}, {3}, \emptyset } )

Set Operations

• A non-empty set contains at least one element. The singleton set contains only one element, e.g., ( {1} ).
• The universal set collects all elements relevant to a problem, denoted as ( S ). The complement of a set ( P ), denoted as ( P' ), consists of elements in ( S ) but not in ( P ).
• De Morgan’s Laws:
  • ( (P \cup Q)' = P' \cap Q' )
  • ( (P \cap Q)' = P' \cup Q' )

Set Membership and Subsets

• If an element belongs to a set ( A ), it is denoted as ( a \in A ). Conversely, ( 3 \notin B ) means 3 is not an element of set ( B ).
• Set ( P ) is a subset of set ( Q ) (denoted ( P \subseteq Q )) if every element in ( P ) is also in ( Q ). A proper subset meets the same condition but is not equal to ( Q ).
• The empty set, denoted as ( \emptyset ) or ( {} ), contains no elements. It serves as a subset of any set.

Subsets Enumeration

• A set with ( n ) elements has ( 2^n ) total subsets and ( 2^n - 1 ) proper subsets. For example, a set with 3 elements has ( 2^3 = 8 ) subsets and ( 7 ) proper subsets.

Intersection and Union of Sets

• The intersection ( A \cap B ) includes elements common to both sets. For example, if ( A = {2, 3, 6, 10, 15} ) and ( B = {3, 6, 15, 18, 21, 24} ), then ( A \cap B = {3, 6, 15} ).
• The union ( A \cup B ) combines all elements from both sets without duplication.

Number Representation

• Natural numbers ( N = {1, 2, 3, \ldots} ) and whole numbers ( W = {0, 1, 2, 3, \ldots} ) can be defined without listing all members, especially for infinite sets.