# Proofs/ counterexamples

• Oct 1st 2011, 07:59 AM
Christyh101
Proofs/ counterexamples
Find a counterexample to show that the converse of each conditional is false.

1. If x=-5, then x^2=25.

2. If two angles are adjacent, then they share a common vertex.

• Oct 1st 2011, 08:11 AM
Plato
Re: Proofs/ counterexamples
Quote:

Originally Posted by Christyh101
Find a counterexample to show that the converse of each conditional is false.

1. If x=-5, then x^2=25.

The converse is $\displaystyle \text{If }x^2=25\text{ then }x=-5~.$
You surely are able to supply a counter-example.
• Oct 1st 2011, 09:54 AM
Soroban
Re: Proofs/ counterexamples
Hello, Christyh101!

Do you know what adjacent angles are?
Then you can surely come up with a counterexample . . .

Quote:

Find a counterexample to show that the converse of the conditional is false.

2. If two angles are adjacent, then they share a common vertex.

The converse is:
. . If two angles share a common vertex, then they are adjacent.

Adjacent angles are angles that share a common side.

The converse is false.
We can have two angles with a common vertex,
. . which do not share a common side.

Code:

      |..../       |.../       |A./       |./       |/       *---------       \.B.....         \.....         \...           \.
• Oct 1st 2011, 10:29 AM
Deveno
Re: Proofs/ counterexamples
in general (and this is worth thinking about now) when you have a statement like "if A, then B" it means that A is a stronger statement than B (and is true for fewer things).

for example, x = -5 tells us everything we need to know about x. only one x will make this statement true, namely, -5.

by contrast, x^2 = 25 only tells us "something" about x, that its square is 25. this happens to be true for two values of x, one is -5, another is 5.

similarly, A and B are adjacent angles, tells us two pieces of information: they share a vertex, and a common side.

this is clearly stronger than: A and B share a common vertex, which is only one piece of information (weaker).

one thing you should have clear in your mind, always: if A is stronger than B, we may not be able to derive A from B.

IF we can, then A and B are "the same strength", that is, they are equivalent statements (true for the same things).

so:

A implies B

A is stronger, true for fewer (or the same number of) things, more restrictive
B is weaker, true for more (or the same number of) things, more inclusive

you will come upon this "shape of thought" in many areas: logical thinking, mathematics (of many kinds), action and consequence. learn it well.