Suppose that you are playing a geussing game with a friend. You have both agreed that you will guess an animal.

Your friend thinks of an animal and you try to guess what it is.

Let's assume, for the sake of this lesson, that your friend always tells the truth.

He tells you that the animal that he is thinking of is a duck-billed animal.

If we call the set of all animals **C**, and the set of all duck-billed animals **A**, then you friend has told you that the animal he is thinking of is in **A**.

If x is the unkown animal, then the statement:

"x is a duck-billed animal"

is true.

That is: A is true.

We represent this fact graphically thus:

Note that because all animals are either duck-billed or not, the statement:

"x is not a duck-billed animal"

is false.

That is: (not A) is false.

We represent this as:

We can combine these two equivalent statements into one picture like this:

One the other hand, suppose that he tells you:

"The animal is not duck-billed."

That is: A is false; equivalently: (not A) is true.