Discrete Math - Sets and Complements I have the following problem: List the elements of the set $\overline{A\cap B}\cup C$, where $\overline{X}$ denotes the complement of an arbitrary set $X$ and $U$ denotes the universe under consideration. The considered sets are as follows:


*

*$U = \{1,2,3,\ldots,10\}$

*$A = \{1,4,7,10\}$

*$B = \{1,2,3,4,5\}$

*$C = \{2,4,6,8\}$


I believe I have the answer but not too sure. Here's what I came up with: 
$$
\overline{A\cap B}\cup C = \{2,3,5,6,7,8,10\}.
$$
Is this answer correct and more so written correctly? 
Thanks in advance. 
 A: Your goal is to determine what $\overline{A\cap B} \cup C$ is. The most sensible thing to do is approach it in a very piecemeal fashion:


*

*Determine what $A\cap B$ is.

*Determine what $\overline{A\cap B}$ is.

*Determine what $\overline{A\cap B} \cup C$ is.





*

*$A\cap B = \{1,4\}$

*$\overline{A\cap B} = U\setminus\{1,4\} = \{2,3,5,6,7,8,9,10\}$

*$\overline{A\cap B}\cup C = U\setminus\{1\} = \{2,3,4,5,6,7,8,9,10\}$


Does that all make sense?
A: With $U = \{1,2,3,...,10\},
A = \{1,4,7,10\},
B = \{1,2,3,4,5\},
C = \{2,4,6,8\}$
The question asks to find $(A\cap B)'\cup C$, where $'$ denotes complement.
We can go about this different ways.  First way would be to do it step by step how it is currently written, following a sort of "order of operations" kind of feel.
$$A\cap B = \{1,4\}\\
(A\cap B)' = \{2,3,5,6,7,8,9,10\}\\
(A\cap B)'\cup C = \{2,3,4,5,6,7,8,9,10\}$$
Another good idea would be to algebraically modify the representation to make it look easier to work with.
$$(A\cap B)'\cup C = A'\cup B' \cup C\\
A' = \{2,3,5,6,8,9\}\\
B' = \{6,7,8,9,10\}\\
A'\cup B' \cup C = \{2,3,4,5,6,7,8,9,10\}$$
