# Math Help - is this argument valid?

1. ## is this argument valid?

some dogs hate cats
all dogs have fleas
frodo has fleas and hates cats
----------------------------
therefore Frodo is a dog

is it valid ? and why?

the explanation should be something like "if P then Q "............

2. No, because "some dogs hate cats" doesn't exclude other species from the hating of cats, and "all dogs have fleas", doesn't exclude others from having fleas, so while Frodo is likely a dog, it's not certain.

3. Hello, ana18081

It is not valid.
I have a Venn diagram proof.

Some dogs hate cats
All dogs have fleas
Frodo has fleas and hates cats
----------------------------
Therefore: Frodo is a dog

Is it valid? Why or why not?

"All dogs have fleas."

There is a set of all dogs.
There is a set of all things that have fleas.
. . $\{\text{dogs}\}\:\subset\: \{\text{have fleas}\}$

The Venn diagram looks like this . . .
Code:
      * - - - - - - - - - - - - - - - - - *
| Has fleas                         |
|       * - - - - - - - - - *       |
|       |                   |       |
|       |       Dogs        |       |
|       |                   |       |
|       |                   |       |
|       * - - - - - - - - - *       |
|                                   |
* - - - - - - - - - - - - - - - - - *

"Some dogs hate cats."

There is a set of all things that hate cats.

The set of "Dogs" and the set of "Hate Cats" do intersect.

The Venn diagram looks like this:
Code:
      * - - - - - - - - - - - - - - - - - *
| Has fleas                         |
|       * - - - - - - - - - *       |
|       | Dogs              |       |
|       |       * - - - - - + - - - + - - - *
|       |       |           |       |       |
|       |       |     X     |       |       |
|       * - - - + - - - - - *       |       |
|               |                   |       |
* - - - - - - - + - - - - - - - - - *       |
|                Hates cats |
* - - - - - - - - - - - - - *
There is at least one creature which is a dog and hates cats.

"Frodo has fleas and hates cats."

He could be in the section with the "X"
. . but he also could be in the shaded region.
Code:
      * - - - - - - - - - - - - - - - - - *
| Has fleas                         |
|       * - - - - - - - - - *       |
|       | Dogs              |       |
|       |       * - - - - - + - - - + - - - *
|       |       |           |///////|       |
|       |       |     X     |///////|       |
|       * - - - + - - - - - *///////|       |
|               |///////////////////|       |
* - - - - - - - + - - - - - - - - - *       |
|                Hates cats |
* - - - - - - - - - - - - - *

Therefore, we can not conclude that Frodo is a dog.

4. Originally Posted by ana1808

some dogs hate cats
all dogs have fleas
frodo has fleas and hates cats
----------------------------
therefore Frodo is a dog

is it valid ? and why?

the explanation should be something like "if P then Q "............
A single counter example will demonstrate that the argument is invalid, so the possibility of existence of a flea infested cat hating hobbit named frodo will prove the argument invalid.

CB