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i tried using the notation to make it all separate and then see if i can put all the solution i get from each a,b, c together but i couldn't understand what i am doing wrong. if anyone could help with understanding this will be great, I have midterm coming up soon and i need to understand how these works. ven diagram

work work
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I love how you care about the steps more than the solution! These kind of problems require normally require you to 'Divide and Conquer'. You know some set operations like $A\cup B$ or $A \cap B$ or $A - B%$. I'd suggest drawing that diagram many times and color each of these possibilities. That is all 12 of these:

\begin{align*} A &\cup B & A &\cap B & A &- B & B &- A \\ B &\cup C & B &\cap C & B &- C & C &- B \\ C &\cup A & C &\cap A & C &- A & A &- C \end{align*}

It is a long process but doing this will get you a good idea of the "pieces" you can color. Once that you know which pieces you can color, you can combine these pieces by using the Union $\cup$. Since you probably have more problems I will tell you the solution: $(C-A)\cup (A\cap B$). Good luck!

Fernando Chu
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  • what do you exactly mean by divide and conquer? – work work Oct 10 '19 at 16:03
  • The use of the little pieces. In the case of this problem I divided it into two different pieces $(C-A)$ and $(A \cap B)$. This division into pieces can be done for every problem. – Fernando Chu Oct 10 '19 at 16:06
  • i also have another question regarding this mathematical question i did some work on paper by using these but as i kept going it got more complicated i couldn't post a pic of that since i written it on my paper but this is the question if you could help me with it any further regarding what do i need to use. A = {xy|x, y ∈ Z, xy = 4} B = {23x+y|x, y ∈ Z, −4 ≤ 2x ≤ 2, xy = 12} it asks for A cordiality and B for how many subsets. i tried looking on youtube there is nothing similar or i even checked my lecture notes negative. – work work Oct 11 '19 at 00:17
  • I'll help you with $A$, $B$ is similar. First, tell me what have you tried for solving it. Also, just in case you didn't know, you should mark as an answer a post that has solved your question. – Fernando Chu Oct 11 '19 at 00:44
  • i tried using the notation of z since z is the set of all integers, then i dont understand the x power of y, and since xy = 4 so it should be like xy e z+ i guess – work work Oct 11 '19 at 07:22
  • if the cardinlity of set A is 4 then does it mean = {xy| x, is one , y ∈ Z, two, xy three = 4 is four? A = {xy |x, y ∈ Z, xy = 4} – work work Oct 11 '19 at 08:39
  • The cardinality of $A$ is the amount of elements $A$ has. Your definition of $A$ asks: how many integers $x$, $y$ are such that $xy=4$? (I'm assuming you mean $A={(x,y) \mid x, y \in \mathbb{Z}, xy=4 }$ right?) – Fernando Chu Oct 11 '19 at 13:29
  • yes except that the first {xy} is {x^y} like the y is on the top of the x as a power or whatever it is calleed. – work work Oct 11 '19 at 15:52
  • Ok, first you have to find all the elements $x, y \in \mathbb{Z}$ such that $xy=4$. Tell me which are them. – Fernando Chu Oct 11 '19 at 16:13
  • i got the answer for a i guess i got the x | y as x values or elements as -1,1,4,-4,2,2 and the y are -4, 4, 1, -1, 2, -2 and i multiply ed each number by each other like -1 with -4 and go on, but i am stuck on question B – work work Oct 11 '19 at 18:50
  • Do the same thing. First check with numbers satisfie $xy=12$, then, check which of those satify $-4\leq 2x \leq 2 $ – Fernando Chu Oct 11 '19 at 19:38
  • yes, so i got 1, -1, -2 for x and 12, -12, -6 for y so this is all the steps basically? – work work Oct 11 '19 at 19:49
  • for this question it asks if A ∩ B = A and C ⊆ A then what is the (A − B) ∪ (C ∪ B) i guessed that the answer is C by looking at the question but i dont know how to prove it. – work work Oct 11 '19 at 19:54