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# 1.3 Equality

1.34   Notation (.) Let and be (names of) objects. I write

to mean that and are names for the same object. I will not make a distinction between an object and its name.

 (1.35)

We describe this property by saying that equality is reflexive.

 (1.36)

We describe this property by saying that equality is symmetric.

 (1.37)

We describe this property by saying that equality is transitive.

Let be a proposition involving the object . Let be a proposition obtained by replacing any or all occurrences of in by . Then . We call this property of equality the substitution property.

1.38   Examples. Suppose that are integers, and . Then

and

and

We will frequently make statements like

The justification for this is

Hence, if , then by the substitution property,

1.39   Warning. Because we are using a vague notion of proposition, the substitution property of equality as stated is not precisely true. For example, although
 (1.40)

and
 (1.41)

are both true, the result of substituting the in the second equation by yields

which is false.

The proper conclusion that follows from (1.40) and (1.41) is

(The use of parentheses is discussed in Remark 2.50.)

1.42   Notation () Let be objects. We write
 (1.43)

as an abbreviation for

If (1.43) is true, then by several applications of transitivity, we conclude that

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