Download Syllabus for MAT 085 - Seattle Central College

MAT 220
Johns
Name
New Truths from Old Truths?
Suppose we start with a known true statement. If we tweek it a little, can we create a new
true statement? Or will we create false ones?
1. For this exercise, we must always first express our true statement as a conditional
statement in the form of If/Then. If our starting truth is not in this form, we need to
carefully get it into this form (an English problem).
For example, let’s start with this truth:
Truth 1: “If x is a rational number, then x2 is a rational number.”
One way of tweeking Truth 1 to get a new statement is to switch the If/Then parts. We
call the resulting statement the converse of Truth 1.
Converse of Truth 1: If
Is the converse true?
, then
If not, find a counterexample.
Another way of tweeking Truth 1 to get a new statement is to negate the If/Then parts.
We call the result the inverse of Truth 1.
Inverse of Truth 1: If x is not a rational number, then x2 is not a rational number.
Is the inverse true? Explain.
Last but least, we can tweek Truth 1 by negating AND switching the If/Then parts.
We call this new statement the contrapositive of Truth 1.
Contrapositive of Truth: If
, then
Is the contrapositive true? Explain.
2. Let’s start with another truth:
Truth 2: The sum of two odd numbers is not an odd number.
First we need to write Truth 2 in the If/Then form:
Truth 2:
If
, then
Now write down the converse, inverse, and contrapositive of Truth 1:
Converse: If
, then
Inverse:
Contrapositive:
Which of these new statements are false?
Which appear to be true?
We should see a pattern here. If the original statement is known to be true, then the
is also true. But the
and
are false.
3. Let’s take a more complicated example. Using independent means, we can prove
Theorem 4 in Chapter 1:
Theorem 4: A homogeneous linear system with more unknowns than equations always
has a nontrivial solution.
Write this theorem in a conditional format. (Note: you do not need the word “always” in
your second part!)
Theorem 4:
If
Then
State the converse of Theorem 4 and prove that it is false.
Converse:
If
Then
State the inverse of Theorem 4 and prove it is false.
Inverse:
If a homogeneous linear system does not have more unknowns than equations,
Then the system might not have a non-trivial solution.
State the contrapositive of Theorem 4:
Contrapositive:
If
Then
Even though we don’t know how to prove its truth, does the contrapositive seem true?
EXTRA PRACTICE
Write the following truths in conditional format. Then write out their converses,
inverses, and contrapositives.
a) A number divisible by 6 is divisible by 2.
b) The product of two even numbers is even.
c) Every positive number x greater than one is smaller than its square x2.
(Note: the negation of “then every x” is not “then no x”; it is “then at least one x is not”!