# Please, help to simplify set theory expression

I need to simplify set theory expression: $$(\bar{B} \cap C) \cup \bar{D} \cup ((B \cup \bar{C}) \cap D) \cup (A \cup \bar{C} )$$
$$\bar{B}$$ means not B

I understand how to solve it graphically with Euler-Venn diagram. But I got stuck with solving it analytically using standard operators such as union, intersection, difference, complement of set.

I will be so grateful if you provide your solution step by step with used properties of algebraic structure like associative and commutative laws and etc.

• What is $\overline B$? Do you mean complement of $B$? – Dbchatto67 Mar 19 at 14:44
• @Dbchatto67 It means not B – Lord of Programs Mar 19 at 16:03
• What I got after simplification is that $A \cup B \cup C \cup {\overline {D}}.$ I think no further simplification can be made. – Dbchatto67 Mar 19 at 16:14
• It may be universum or empty set in the end, ideally. Could you describe, your simplification step by step, please? – Lord of Programs Mar 19 at 16:24
• I got U as the answer . – ADITYA PRAKASH Mar 19 at 19:09

I am not using the TeX codes , sorry . I hope you can understand this handwritten version.

• Thank you, it is exactly, what I need!) – Lord of Programs Mar 20 at 1:56