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I posted a similar question here, however I have another question regarding Venn diagrams and logic circuits...

In this problem:

$$(A+B)(B+C)$$

Wouldn't the Venn diagram look something like this?

enter image description here

Because it simplifies to $B + AC$? And you can either have the $B$ region, the $AC$ region, or both?

However, apparently it's supposed to look something like this:

enter image description here

Which I don't understand... Because by the logic that you have $AB$ and $BC$, couldn't you justify saying that you can also shade in $A$, or shade in $C$ (by themselves)?

OR, does saying $A$ mean you shade in EVERYTHING that contains $A$? And only don't shade in the intersection if you had something that said $A * \overline{B}$?

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  • $\begingroup$ I've edited your title and first paragraph, because this isn't a repost of your earlier question: it's a question about a different formula. $\endgroup$ – David Richerby Aug 4 '17 at 18:54
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The misunderstanding here is the same as in your previous question. If $B$ is true, the formula is true whatever the values of $A$ and $C$ are. This means that the whole of the $B$ circle needs to be shaded. Shading just the part at the top corresponds to saying "The formula is true if $B$ is true and $A$ and $C$ are both false." That is certainly true, but it's not the whole story.

For example, your proposed shading says that the formula is false if $A$ is true, $B$ is true and $C$ is false. But the formula is true in that case! You must shade every region of the diagram that makes the formula true. In this case, that means every region in the $B$ circle, and also the intersection of the $A$ and $C$ circles. (Corresponding to, "The formula is true if $B$ is true, or if $A$ and $C$ are true, or both.")

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