# MandysNotes

Sunday, 06 April 2014 00:00

## Truth Values for Disjunction

By  Gideon
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Suppose your friend tells you that he is thinking of an animal and that this animal is either duck-billed, or a mammal.

In other words he is telling you that $$A \cup B$$ is true.

Assume that your friend is telling the truth.

He is telling you that whatever animal he is thinking of, it will be somewhere in the green area in the figure below.

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

This means that $$A$$ is true.

Now he may tell you that the animal is also a mammal, this means that $$B$$ is true; or he may tell you that the animal is not a mammal, this means that $$B$$ is false.

But either way, as long as $$A$$ is true, $$A \cup B$$ is true.

That is, even if the animal is duck-billed, but not a mammal, the statement:

"The animal is either duck-billed or a mammal" is true.

We can represent the situation where $$A$$ is true, but $$B$$ is false, like this:

Now suppose that your friend first tells you that the animal is either duck-billed or a mammal:

$$A \cup B$$ is true,

and then he tells you that the animal is a mammal, that is:

$$B$$ is true.

Now the animal may also be duck-billed, or it may not be duck-billed,

that is $$A$$ may be true, or $$A$$ may be false, but either way:

$$A \cup B$$ is true.

Finally, consider that your friend tells you that $$A \cup B$$ is false, that is he tells you that the animal is neither duck-billed, nor a mammal.

In that case $$A \cup B$$ is false.

Note that this is the only way to make $$A \cup B$$ false: both $$A$$ and $$B$$ must be false.

This means that whatever animal your friend is thinking about, it cannot be in the red area in the figure below.

We can express all this with a truth table for $$A \cup B.$$

Note that $$A \cup B$$ is true unless both $$A$$ and $$B$$ are false.