If your friend tells you that the animal he is thinking of is duck-billed, and is a mammal, then he is telling you that \(A\) is true, and that \(B\) is true, and hence that your guess was right, that \(A \cap B\) is true.

On the other hand, if your friend tells you that the animal is duck-billed, but is not a mammal, then he is telling you that \(A\) is true, but that \(B\) is false.

We can represent this by the following figure:

Because the animal that your friend is thinking of will with certainty not be in the red area, \( A \cap B \) is clearly false.

Similarly, if he tells you that the animal is not duck-biled, but is a mammal, he is telling you that \(A\) is false, but that \(B\) is true.

We represent this as:

Once again we see that \( A \cap B\) is in the red area of this diagram, and so must be false.

Finally if your friend tells you that the animal is not duck-billed and not a mammal, then he is telling you that \(A\) is false, and that \(B\) is false.

In this case, \(A \cap B\) is clearly false.

We can represent all these truth values in a truth table for \(A \cap B.\)

Note that \(A \cap B\) is only true when both \(A\) and \(B\) are true, otherwise it is false.