A.4 Properties of Equality 611Notice that neither of these theorems claims that the statements are all
true, only that they are equivalent. That means that, if any one of them is
true, all the others are true as well. If any one of them is false, all of them are
false.
A.4 Properties of Equality
Students sometimes ask how we justify the familiar manipulation of equality
(the "equals sign") in proofs. In mathematics, as elsewhere, the term "equals"
is understood to mean "is the same as." Thus, for example, when we write
{8,5,4} = {5,8,5,4}^4
we are indicating that the expressions on either side of the equals sign represent
the same set; the two sets are indistinguishable as sets. Of course, there is
something distinguishable about the two sides of the equation, or else our eyes
could not perceive that the equation has two sides, and we would have no reason
to ask whether they are the same. Perhaps a more mature understanding of
"= " is that it means "is, for our present purposes, the same as."
RULES OF EQUALITYThe familiar rules of equality used most frequently in proofs are(1) The reflexive property: '\Ix, x = x.
(2) The symmetric property: '\Ix, y, x = y =} y = x.(3) The transitive property: '\Ix, y, z, (x = y and y = z) =} x = z.
(4) The replacement property: In any context in which x = y, either x
or y can be replaced by the other whenever it occurs.
The replacement property has subtle applications. For example, the familiar
assertions
( 1) if x = y, then V z, x + z = y + z, and
(2) if a= band c = d, then a+ c = b + d
are justified by the replacement property of equality. These principles are more
commonly stated, "when equals are added to equals, the results are equal."
They are not axioms about addition, but are properties of equality.
- See Appendix B.l for a discussion of "set" notation.