The problem is as follows:
To participate in an edition of The Price is Right $60$ people showed up. The producer of the show collected the following information:
$10$ women have black eyes $16$ women don't have black eyes $14$ women do not have brown eyes $10$ men do not have brown or black eyes
Based on information from the show's producer. How many men have brown or black eyes?
The alternatives in my book are as follows:
$\begin{array}{ll} 1.&\textrm{24 men}\\ 2.&\textrm{22 men}\\ 3.&\textrm{18 men}\\ 4.&\textrm{28 men}\\ 5.&\textrm{20 men}\\ \end{array}$
What I attempted to do is summarized in the table from below which showcases my findings. I must indicate that I arranged the characters of the story in two groups. One for women and the other for men. One column indicates $F$ for false of those who do not have a certain characteristic, i.e. A man who does not have black eyes would be in the F column below black eyes, a woman who has black eyes will lie below T in the black eyes column.
Hence this is represented as:
Where $x+y=10$
Adding all of these yields:
$14+10+16 +x+y+a+b=60$
Since it mentions $x+y=10$ and they want to know what is a+b then this would mean:
$a+b=60-50=10$
However this is not within the alternatives.
In my effort to better represent this situation and avoid counting twice I made the diagram from below:
This different representation using an Euler diagram shows what I thought, the set of those who do not have brown eyes must account for those who have black eyes.
There is no interesection in any of these sets as there cannot be someone who has brown and black eyes at the same time. Thus I concluded what they want is the number of men who have brown eyes and those who have black eyes.
This part I require clarification, what is the meaning of OR translated in set theory language?. Does it mean is it a Union?. Is it $\cup$?. Can someone help me with this part?.
Returning to my attempt I then thought that the number $14$ of those who do not have brown eyes will include those who have black eyes therefore.
Adding those who have black eyes with those who have brown eyes with those who dont have brown neither black in the women side. With those on the men side who dont have neither black or brown with those who have brown and those who have black will equate to $60$. This translated into an equation would be as this:
14+16+10+u=60
Assuming $u=\textrm{unknown males with brown with those with black eyes}$
This would become into:
$u=20\,men$
But this is not the right answer. Why?.
Did I made a mistake or something?. Because of this reason I need a very detailed help with this.
Please an answer must include a Venn or Euler diagram strategy or using a table. To better aid the understanding of this problem. I have tried all sorts, but I don't seem to get the right answer.
Can someone include also the representation of set notation using algebraic notation accompanied with an Euler or Table diagram?.
You are given:
Assuming no one have heterochromia, each person only has one single eye color.
Hence either they have black eyes, or they do not.
From the first two conditions we see that there are $26$ women. Thus there are $34$ men.
Now $10$ men do not have brown or black eyes, and we are to find the number of men with brown or black eyes, which is the complement of those $10$ men, so there are $24$ of them.
The major mistake in your answer is the equation
$$14+10+16+x+y+a+b=60$$
Since the $14$ women without brown eyes and those $16$ women without black eyes are not mutually exclusive, you can't just add them up; you will overcount.
To answer your side question, OR translates to the union $\cup$ and AND translates to the intersection $\cap$. However these do not come into play in this specific question.