Algebra Know-It-ALL

182 Equations and Inequalities

Are you confused?

Once you know what all these symbols mean, you can read complicated logical statements out loud or

write them down in words. You can even break them up and turn them into “mathematical verse.” For

example, the last logical sentence in the previous paragraph can be written like this.

For all a, b, and c:

If

a is smaller than or equal to b,

and

b is smaller than or equal to c,

then

a is smaller than or equal to c.

Here’s a challenge!

Is logical implication an equivalence relation?

Solution

No. To see why, let’s make up three statements and give them logical names:

• I’m thinking of a natural number =Tn.


• I’m thinking of a rational number =Tq.


• I’m thinking of a real number =Tr.

Now let’s check out the symmetric property. From our knowledge of how the sets of naturals, rationals,

and reals (N, Q,andR) are related, we know that

Tn⇒Tq

and

Tq⇒Tr

These statements translate to the following mathematical verses.

If

I’m thinking of a natural number,

then

I’m thinking of a rational number.

If

I’m thinking of a rational number,

then

I’m thinking of a real number.