Whereas the ‘$\,=\,$’ sign gives a way to compare mathematical expressions,
the idea of equivalence gives a way to compare mathematical sentences.
To motivate the idea of equivalence, consider these two mathematical sentences:
$2x-3 = 0$ | and | $\displaystyle x = \frac{3}{2}$ |
They certainly look different.
But, no matter what value is chosen for the variable $\,x\,$,
these two sentences always have the same truth values.
Indeed,
$\,2x - 3 = 0\,$
is true only when
$\,x\,$ is
$\,\frac{3}{2}\,$, and false otherwise.
Also,
$\, x = \frac{3}{2}\,$
is true only when
$\,x\,$
is
$\,\frac{3}{2}\,$, and false otherwise.
When two mathematical sentences always have the same truth values,
then they can be used interchangeably,
and you can use whichever sentence is easiest for a given situation.
The mathematical verb used to compare the truth values of sentences is:
‘is equivalent to’.
Be careful, because equal and equivalent have totally different uses in mathematics!
You compare expressions using ‘equal’.
(Numbers can be equal, sets can be equal.)
You compare sentences using ‘equivalent’.
(Equations can be equivalent, inequalities can be equivalent.)
To make the idea of ‘equivalence of sentences’ precise,
we must first talk about
connectives and compound sentences.
Mathematicians frequently take ‘little’ things
and connect them into ‘bigger’ things,
using appropriate connectives.
Once connected up, the result is often referred to as a
compound thing:
$$
\cssId{s32}{\overset{\text{compound thing}}{\overbrace{\text{thing1 }\ldots\text{ connected to }\ldots\text{ thing2}}}}
$$
There are different types of connectives, depending on what is being connected.
Numbers can be ‘connected’ to get a new number.
The four most common connectives for numbers are:
Sets can be ‘connected’ to get a new set:
union and intersection are two common set connectives.
Sentences can be ‘connected’ to get a new sentence:
the mathematical words ‘and’, ‘or’, and ‘is equivalent to’
are sentence connectives.
For example:
if $\,A\,$ is a sentence and $\,B\,$ is a sentence,
then $\,A\text{ and }B\,$ is a compound sentence.
The truth of this compound sentence depends upon the truth of the subsentences
$\,A\,$ and $\,B\,$.
A truth table shows how the truth values of a compound sentence relate to the
truth values of its subsentences.
Here are the definitions of the mathematical words and,
or, and
is equivalent to:
$\,A\,$ | $\,B\,$ | $\,A\text{ and }B\,$ | $\,A\text{ or }B\,$ | $\,A\text{ is equivalent to }B\,$ |
T | T | T | T | T |
T | F | F | T | F |
F | T | F | T | F |
F | F | F | F | T |
Some important observations from the truth table:
The idea of mathematical equivalence is so important that there are many ways to say the same thing.
The following four mathematical sentences are equivalent:
if one is true, they all are true;
if one is false, they all are false.
These four sentences are completely interchangeable!
If | $\,A\,$ is true | and | $\,B\,$ is false, | then the sentence | $\,A\text{ and }B\,$ | is false. |
If | $\,A\,$ is false | and | $\,B\,$ is true, | then the sentence | $\,A\text{ or }B\,$ | is true. |
If | $\,A\,$ is false | and | $\,B\,$ is false, | then the sentence | $\,A\text{ iff }B\,$ | is true. |
If | $\,A\,$ is true | and | $\,B\,$ is false, | then the sentence | $\,A\text{ is equivalent to }B\,$ | is false. |
If | $\,A\,$ is true | and | $\,B\,$ is true, | then the sentence | $\,A\text{ if and only if }B\,$ | is true. |