Solve Math Sets Homework: (C-\negA)\cup(A\capB)\cup(C\capB)=C

[Renogen]

Homework Statement

(C-[tex]\neg[/tex]A)[tex]\cup[/tex](A[tex]\cap[/tex]B)[tex]\cup[/tex](C[tex]\cap[/tex]B) = C

Homework Equations

No idea what to be put here.

The Attempt at a Solution

(C[tex]\cap[/tex]A)[tex]\cup[/tex](A[tex]\cap[/tex]B)[tex]\cup[/tex](C[tex]\cap[/tex]B) = C
{[(C[tex]\cap[/tex]A)[tex]\cup[/tex]A][tex]\cap[/tex][(C[tex]\cap[/tex]A)[tex]\cup[/tex]B]}[tex]\cup[/tex](C[tex]\cap[/tex]B)=C
A[tex]\cap[/tex][(C[tex]\cap[/tex]A)[tex]\cup[/tex]B][tex]\cup[/tex](C[tex]\cap[/tex]B)=C

got stuck after this, help needed, thanks.

 

[KataKoniK]

Hello there! It looks like you have made some progress in solving this problem. Let's break down the steps you have taken so far:

1. You correctly applied the distributive property to expand the expression. This is a good first step in simplifying the equation.

2. Next, you simplified the first term by using the identity property of intersection. This is also correct.

3. However, in the second term, you used the union property instead of the intersection property. Remember that in this step, we are only simplifying the expression and not changing the overall structure. So, the second term should be written as (A\capB) instead of (A\cupB).

4. Applying the identity property of intersection again, we can simplify the second term to just (A\capB).

5. The third term, (C\capB), remains unchanged since there are no further simplifications we can make.

6. Finally, we can use the distributive property in reverse to combine the three terms into one, resulting in (C\capB)\cup(A\capB)\cup(C\capB).

7. Using the idempotent property of union, we can simplify this to just (C\capB), which is equivalent to the original expression of C.

I hope this helps! Let me know if you have any further questions. Good luck with your studies!