Math Notes – Singleton Sets

A singleton set is a set that contains only one member.  Some theorems that apply to singleton sets are not generally true for all sets, so the knowing that a set is a singleton set can be important in proving theorems.

Suppose that the element ‘a’ is a member of the universal set U.  The set {a} is a singleton set.  A singleton set represents a set such that all members of the set equal the same element.  Thus, a singleton set may be defined as the following axiom.

Axiom 1: {a} = {x|x=a:x}.

A singleton set is identified by the member associated with the set.  If it can be determined that an element is a member of a singleton set, then the element must be the same as the element associated with the singleton set.  A simple result of this theorem is that an element is a member of the singleton set associated with the set.

# Theorem 1: b Î {a} º b = a

Corollary: x Î {x}

For each property of an element, there should be a corresponding property of the singleton set associated with the element.

Theorem 2: a Î P º {a} Í P

Some properties of singleton sets are not generally true for all sets.  Theorem 3 is an equivalence expression for singleton sets, but only an implication if generalized to all sets.

Theorem 3: {a} Í P Ú {a} Í Q º {a} Í (P È Q)

A singleton set is a member of a set or the set’s complement.  This property is not generally true of sets with two or more elements.

Theorem 4: {x} Í B Ú {x} Í ~B

Problems

Construct a Venn diagram that demonstrates why the expression in Theorem 3 is only an implication when generalized to include all sets.

Suppose

P = {1,2,3}

Q = {4,5,6}

Construct a set R that is not a subset of either set P or set Q, but is a subset of the union of P and Q.

References:

Fejer, Peter and Simovici, Dan A., 1991, Mathematical Foundations of Computer Science, New York, NY: Springer-Verlag

Ganong, Rick 1999, Course notes and comments from M1090 and M2090 math and logic courses, http://www.math.yorku.ca/Who/Faculty/Ganong

Gries, David and Schneider, Fred B., 1993, A Logical Approach to Discrete Math, New York, NY: Springer-Verlag

Hu Sze-Tsen, 1963, Elements of Modern Algebra, San Francisco: Holden-Day, Inc.