Examples and Explanations of Basic Properties of Equality

Date: 07/31/2004 at 15:02:12
From: Alyssa
Subject: identity and equality properties

Can you put these properties in simpler terms?  I have no clue what 
they are trying to say.

Substitution property of equality: for any numbers a and b, if a = b, 
then a may be replaced by b in any expression.

Symmetric property of equality: for any numbers a and b, if a = b, 
then b = a.

Transitive property of equality: for any numbers a, b and c, if a = b 
and b = c then a = c.

I tried substituting numbers for the variables, but it didn't make it 
any clearer.

Date: 07/31/2004 at 22:39:51
From: Doctor Peterson
Subject: Re: identity and equality properties

Hi, Alyssa.

These are all so basic that it can be hard to see why we bother saying
them!  What mathematicians try to do is to boil down our reasoning to
the simplest clear statements we can make, so that we can prove
everything we say on the basis of those "axioms".  Let's look at what
they mean.

  Substitution:

Suppose I know that "big" is just another word for "large".  That 
means that anywhere I read the word "large", I can replace it with the
word "big".  That's obvious.  (Though in language, sometimes things
get a lot more tricky than in math, since words can have more than one
meaning!)

The same thing works in math.  If I know that two variables both refer
to the same number (that's what "equal" means), then anything I say
about one of them is also true of the other.  If I know

  a = b

and you give me the expression

  3a^2 - 5

(that is, 3 times the square of a, minus 5), then I know I can replace
"a" with "b" in that expression, and

  3b^2 - 5

will always have exactly the same value.  If, for example, a and b are 
both equal to 4, then regardless of whether I call it a or b, I get 
the same answer (43) when I evaluate the expression.


  Symmetric property:

The word "symmetry" means that if we reverse something, it still looks
the same.  For example, the word "wow" is symmetrical, because if I
look at it in a mirror so that it is reversed, it looks the same.

Equality is "symmetrical" in a similar sense: if I have an equation

  a = b

then if I read it backwards, as

  b = a

it is still true.  It's obvious that if this is equal to that, then 
that is also equal to this.  But we call attention to this because 
there are other "relations" we can define that are NOT symmetrical. 
For example, suppose instead of "=" we were talking about ">".  Then 
if we take the fact

  1 < 3

which is true, and turn it around to get

  3 < 1

it becomes false.  The "less than" relation is NOT symmetrical.

But, knowing that equality IS symmetrical, we know that if we have an 
equation like

  3x - 5 = y

we can rewrite that as

  y = 3x - 5

and it is still true.  That is important in solving equations.


  Transitive property:

The word "transitive" means "going through"; it's related to a 
"transition", which "goes through" from one situation to another.  
Going back to words, since "big" means the same as "large", and 
"large" means the same as "sizeable", we know that "big" means the 
same as "sizeable".

In algebra, we say that if we have three variables a, b, and c, and we 
know that

  a = b  and  b = c

then the three variables are ALL equal to one another, and

  a = c

In simple words, if two things are equal to a third thing, then they 
are equal to one another. As an example, if we know that

  3x - 5 = 2y + 3

and that

  2y + 3 = 5z - 1

then we can conclude that

  3x - 5 = 5z - 1

Using specific numbers in place of the variables is tricky if you want 
to illustrate these properties: since everything is equal, all the 
numbers would have to be the same!  The transitive property, for 
example, says that since we know that 1=1 and 1=1, therefore 1=1.  Not 
very interesting!  It works a lot better to do as I have done above 
and use variables or words as examples.

But try this: suppose I know that my math book weigh the same as my 
history book, and my history book weighs the same as two English 
books.  Then the transitive property tells me that my math book weighs 
the same as two English books.  Does that make it clearer?

If you have any further questions, feel free to write back.

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/