- We have standard names for the sets of rational and irrational numbers: Q and S,
respectively. These sets are disjoint; they have no elements in common. If x is an
element of Q and y is an element of S, thenx is never equal to y. In logical form along
with set notation, we can write
[(x∈Q) & (y∈S)]⇒x≠y
- We can write the statement as “mathematical verse” by reading it out loud and taking
careful note of each symbol. Here’s the logical statement again, for reference.
(∀a,b,c) : [(a≥b) & (b≤c)]⇒ (a=c)
When we break up the statement into parts and write them down on separate lines, we
come up with the following:
For all a,b, and c:
If
a is larger than or equal to b,
and
b is smaller than or equal to c,
then
a is equal to c.
This little poem might be cute, but it doesn’t state a valid mathematical law. Suppose that
a= 5, b= 3, and c= 7. In that case, a is larger than or equal to b and b is smaller than or
equal to c. However, a is not equal to c.
- Our task is to simplify the equation to a form where x appears all by itself on the left
side of the equality symbol, and a plain numeral appears all by itself on the right. Here’s
the equation again, for reference:
x+ 4 = 2 x
We can subtract x from both sides, getting
x+ 4 −x= 2 x−x
which simplifies to
4 =x
We can reverse the order to get the solution in its proper form:
x= 4
The original equation holds true only when x is equal to 4.
Chapter 11 621
