I am currently having trouble understanding these sets.
I am on letter a and I worked out that k is an element of P, where p is composed of positive integers, such that "k-1 is a multiple of 7." No clue what to do with this part.
[Math] List four elements of each of these sets
discrete mathematics
Related Solutions
The notation $A=\{x|P(x)\}$ or equivalently $A=\{x:P(x)\}$ means that the set $A$ contains all $x$ for which the condition $P(x)$ is satisfied.
So your set contains all $x$, such that $x$ is a real number and $x$ satisfies the equation $x^2=1$. In other words, your set contains only the square roots of $1$, which are (as you said) $\pm1$:$$\{x|x\in \mathbb{R},x^2=1\}=\{1,-1\}.$$
A Venn diagram might help
If you are uncertain, or want to be really bullet proof, you can go through all of the cases. $x\in X$ and $Y$... $x\in Y$ but not in $X,$ etc. and enumerate all of the cases. And, as you get more experience you will see that some are irrelevant and don't need to be discussed.
Suppose the proposition is not true. Then there is an element of $x\in X\cap Z$ that is not in $Y\cap Z.$ But that would imply that there is an $x\in X$ that is not in $Y$ which is in contradiction to the statement that $X$ is a subset of $Y.$
Best Answer
$\mathbb{P}$ is weird notation for positive integers. (I would read it as the primes). Anyway, you just need to find four positive integers $k$ such that $k-1$ is a multiple of 7. This means $k$ is one more than a multiple of 7. 7 is a multiple of 7, so $k=8$ is in the set. Just think of a few other multiples of 7 and proceed accordingly.