1.34
Notation (
.)
Let
![$x$](img85.gif)
and
![$y$](img86.gif)
be (names of) objects. I write
to mean that
![$x$](img85.gif)
and
![$y$](img86.gif)
are names for the same object. I will not make a
distinction between an object and its name.
1.38
Examples.
Suppose that
![$x,y$](img149.gif)
are integers, and
![$x=y$](img150.gif)
. Then
and
and
We will frequently make statements like
The justification for this is
Hence, if
![$x=y$](img150.gif)
, 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
![\begin{displaymath}
5=2+3
\end{displaymath}](img157.gif) |
(1.40) |
and
![\begin{displaymath}
5\cdot 4=20
\end{displaymath}](img158.gif) |
(1.41) |
are both true, the result of substituting the
![$5$](img64.gif)
in the second equation by
![$2+3$](img159.gif)
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.)