Confusion on wording of PIE question, "and" vs "or"

Question

The Sanders family has 3 boys and 3 girls. In how many ways can the 6 children be seated in a row of 6 chairs, so that the boys aren't all seated together and the girls aren't all seated together?

It seems to me that the problem is stating that the configurations $\text{BBBGGG}$ and $\text{GGGBBB}$ are illegal, while a configuration such as $\text{BGGGBB}$ is legal, because although all of the girls are together, it is not true that all of the boys are seated together and all of the girls are seated togther, so it is true that all of the boys aren't seated together and all of the girls aren't seated together. However, part of the given solution is this:

There are 4 configurations with the 3 boys together - we could list them as $\text{BBBGGG, GBBBGG, GGBBBG,}$ and $\text{GGGBBB}$... Similarly there are 4 configurations with the 3 girls together. But this does not mean that there are $4+4=8$ illegal configurations! We have double-counted the configurations where both the boys and the girls are together. There are two of these - $\text{BBBGGG}$ and $\text{GGGBBB}$ - so we must subtract 2 from our total. Therefore there are 6 illegal configurations of boys and girls...

I do not understand why the configurations like $\text{GBBBGG}$ are counted as illegal, because it's my understanding that logical $\text{and}$ means that a statement is true if both of the conditions are true. What am I missing, regarding the problem statement?

Answer 1

For a legal configuration, we need that " the boys aren't all seated together and the girls aren't all seated together". For this and statement to hold, we need both parts to hold: (1) The boys must not all be seated together; and (2) the girls must not be all be seated together. If either of these conditions fail, the whole 'and'-statement fails.

So for instance, in the configuration BBGGGB, since the girls are all seated together, (2) fails, and so this is a configuration that is not of interest.

Answer 2

You are right a conjunction, i.e. a statement of the form $P \land Q$ is true if and only if both $P$ and $Q$ are true. So let's look at your case. We have

• $P$ is the statement "all of the boys are not sitting to gather"
• $Q$ is the statement "all of the girls aren't sitting together"

and for a configuration we must have both true.

How do we check that a configuration is legal? We must check that both $P$ and $Q$ are true. For example:

• If the configuration is $BGBGBG$ then clearly all boys are not sitting together ($P$ is true) and all girls are not sitting together ($Q$ is true), so this configuration is legal.
• If the configuration is $BGGGBB$ then $P$ is true since one boy does not sit with the others. But what about $Q$? It is the case that all girls are sitting together, so the statement $Q$ is false. Thus $P \land Q$ is false and the configuration is illegal.

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Here is a more number-focused example: suppose you have the following statements:

• $P$ is the statement "x > 0"
• $Q$ is the statement "x < 2"

What numbers satisfy both $P$ and $Q$? You can see that some chosen number $x$ satisfies $P \land Q$ if and only if $0 < x < 2$.

So what about $x = 3$? Clearly, $P$ holds in this case since $3 > 0$. But $Q$ clearly fails. So would you argue that $P \land Q$ holds?