Union and intersection of given sets (even numbers, primes, multiples of 5) I am trying to compute the intersections and unions of the following sets ...
$A=\{x:x\in \mathbb{N}\ \text{and x is even}\}$
$B=\{x:x\in \mathbb{N}\ \text{and x is prime}\}$
$C=\{x:x\in \mathbb{N}\ \text{and x is a multiple of 5}\}$
a. $A\cap \!\,B=$ {2}
even and prime, if it's even it's divisible by 2, so 2 is the only even and prime number
b. $B\cap \!\,C=$  {5}
multiple of 5 and prime, so 5 is the only choice
c. $A\cup \!\,B=$ {all prime natural numbers and all even natural numbers}
d. $A\cap \!\,(B\cup \!\,C)=$ {all even numbers that are either prime or divisible by 5}
I am not sure about my answers for parts c and d specifically. I understand union and intersection but I am a little unsure about how to actually write out my answers.
 A: Your answer for part c is correct. There are a number of more "formal" or "mathy" ways of writing it. For example, $A\cup B=\{n:n\in\Bbb{N}, 2|n\vee n\in\Bbb{P}\}$. This means, "the set of all $n$ where $n$ is a natural number, and $2$ divides $n$ or $n$ is prime." However, using the symbol $\Bbb{P}$ to represent the prime numbers is not universally accepted. It is perfectly acceptable to use words in certain mathematical statements, so you could say
$$A\cup B=\{n\in\Bbb{N}:n\textrm{ is prime or }n\textrm{ is even}\}.$$
Your answer for part d is also correct. For this you could say
$$A\cap(B\cup C)=\{n\in\Bbb{N}:2|n\textrm{ and }(5|n \textrm{ or }n\textrm{ is prime})\}.$$
In general, a union means you combine the set conditions with an "or" and an intersection combines them with an "and". Also, if a "mathy" description is more cumbersome than a simple one, as is the case with primes, it is usually okay to use English. We know what prime numbers are; you don't need to say
$\{p\in\Bbb{N}:nq=p\wedge q\in\Bbb{N}\Longrightarrow q=1\vee q=p\}$.
A: It's all ok, but you could expand a bit on d); personally, I'd distribute the intersection to understand better what's happening:
$A\cap \!\,(B\cup \!\,C)=(A\cap B)\cup (A\cap C)$
First part, you answered in a.: {2}
Second part: even multiples of 5, so multiples of 10.
All in all, {x:x=2 or x multiple of 10}.
