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.
. . $\displaystyle \{\text{dogs}\}\:\subset\: \{\text{have fleas}\}$
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.