The dual to set theory is what I call “class theory”. Contradiction corresponds to the null set, and tautology corresponds to the universal set.
Material implication (→) corresponds to “is a subset of”. Set theory has a well-known correspondence with logic: negation (¬) corresponds to complement, disjunction (OR, ∨) corresponds to union, and conjunction (AND, ∧) corresponds to intersection. An intersection (∩) of sets is defined as the set whose members are contained in every referenced set.
A union (∪) of sets is the set containing all members of the referenced sets. The complement of a set ( c) is the set of all elements within a particular universe that are not in the set. An unrestricted universal set is not defined because it would lead to contradictions. A universal set (Ω) is defined as having all members within a particular universe. That is paradoxical but not contradictory. The null set (∅) is a unique set defined as having no members. If subset s does not (or cannot) equal S, then it is a proper subset: “s ⊂ S”. A subset is a set whose members are all within another set: “s is a subset of S” is “s ⊆ S”. The notation for “x is an element of set S” is “x ∈ S”. Its properties may also be known or specified, but what is essential to a set is its members, not its properties. To some extent, the original and the dual may be used together.Ī set is defined by its elements or members. Let us begin with the standard approaches to these three topics, and then define duals to each of them. (1) Set theory and logic, (2) number and algebra, and (3) space and time are three foundational topics that each have duals.
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