Definition - An item is a thing.
Definition - A term is a word or a phrase.
Definition - A character is a mark.
Definition - K is a name if and only if K is a term or character that refers to a thing.
Definition - A thing has a name if and only if a term or character refers to it.
It should be noted that if K is a name of thing T, it may not be the only name that refers to thing T.
Definition - An element is a thing, T, contained by something other than T.
Definition - S is a set if and only if each following statement is true:
S may contain an element or elements, or S may contain no elements.
Each element in S is an individually separate, distinct and discrete thing.
Each element in S is unique in S.
Each element in S is represented by a text item that is unique among the text items representing elements in S.
If S is said to contain an element or elements of a given description, and that description is faulty, then S does not contain such elements or such an element.
S does not contain itself.
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Definition - S is an empty set if and only if S is a set, and S does not contain anything.
If set H is a set of butterflies that fly at the speed of sound, set H is surely empty.
Notation - Here is a set containing a pair of elements: {a, b}
Every element in the set except the last element on the right is followed immediately by a comma and then by a space. The elements are enclosed in curly brackets.
Definition - Set S is a non-empty set if and only if S contains an element or elements.
Definition - Set S and set T are equal sets or are equal if and only if S contains each element in T, and T contains each element in S.
Definition - C is a subset of D if and only if C and D are sets and every element in C is also in D.
Definition - C is a proper subset of D if and only if C and D are sets; and, every element in C is also in D; and, there is at least a single element in D that is not in C.
Definition - The union of set T and set S is the set U of elements that are each either in set T or in set S.
Definition - The intersection of set T and set U is the set S of elements that are each in both set T and set U.
Definition - An element E is removed from set S if and only if E is in set S, and S is then redefined to exclude E.
Definition - An element E is inserted in set S if and only if E is not in set S, and E is not identical to any element in S, and S is then redefined to include E.
Definition - An element E is moved from set S to set T if and only if E is removed from set S, and then inserted in set T.
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