Math Notes – Singleton Sets

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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.

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.

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**References:**

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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.