Math from Words

Chapter Two - Sets

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As was said in the introduction, the term *thing* is intended to be thoroughly indefinite. An idea, an object, a sentence, a mark, an action, a description, a time, a location, or anything else can be referred to as a thing. The plural of this term implies that the *things* referred to individually separate, distinct and discrete things. They each have an existence that is independent from each other. A single tomato in a pair of tomatoes exists as independently from the pair of tomatoes as a tomato does among apples.

**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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