Math form Words
Chapter Two - Sets

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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 contained in a collection that is not the collection itself.

The undefined term collection is used here in the sense that it may contain a single thing or more things. Here, it is not required to contain at least a pair of things, as the term is sometimes used. An element contained in a collection may be of any description. Also, a collection may not contain itself. A collection may be an element in another collection, but it may not contain in itself.

Definition - A given thing is a set if and only if each following statement is true:

The given thing has a name. For this definition, its name is S.

S may contain an element or elements.

Every element in S is unique in S.

S does not contain itself.

Definition - S is an empty set if and only if S is a set and S does not contain anything.

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 - 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, or in both set S and set T.

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 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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