# MandysNotes

Sunday, 06 April 2014 00:00

## Truth Values for Conjunction

By  Gideon
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Suppose that your firend is thinking of an animal, and you guess that he is thinking of a platypus.

You say:

"The animal is both duck-billed and a mammal."

In other words, you are saying that $$A \cap B$$ is true.

You are saying that the animal will be in the green section of the figure below.

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.