(AuB)\C=(A\C) u (B\C) Prove or Disprove

[katburns03]

(AuB)\C=(A\C) u (B\C) Prove or Disprove this statement.

 

[Deleted member 4993]

(AuB)\C=(A\C) u (B\C) Prove or Disprove this statement.

Please share your work with us .

If you are stuck at the beginning tell us and we'll start with the definitions → "what does AuB translate to in english?"

You need to read the rules of this forum. Please read the post titled "Read before Posting" at the following URL:

http://www.freemathhelp.com/forum/th...217#post322217

 

[katburns03]

It means the union of. I am not even sure where to begin with this.

 

[pka]

(AuB)\C=(A\C) u (B\C) Prove or Disprove this statement.

This whole question turns on on simple principle in logic:
\(\displaystyle \left( {A \vee B} \right) \wedge \neg C \equiv \left( {A \wedge \neg C} \right) \vee \left( {B \wedge \neg C} \right)\).

 

[HallsofIvy]

(AuB)\C=(A\C) u (B\C) Prove or Disprove this statement.

The standard way to prove "set X= set Y" is to prove first "set X is a subset of set Y" and then "set Y is a subset of set X". The standard way to prove "set X is subset of set Y" is to start "if x is a member of set X" and then use the properties of sets A and B to conclude "then x is a member of set Y".

Here, "X" is \(\displaystyle (A\cup B)\ C\) so "if x is a member of \(\displaystyle (A\cup B)\C\) then x is in \(\displaystyle A\cup B\) but not in C. Since x is in \(\displaystyle A\cup B\) either x is in A or x is in B.

Either
1) x is in A but not in C so x is in A\C and so is in \(\displaystyle (A\C)\cup (B\C)\).

2) x is in B but not in C so x is in B\C and so is in \(\displaystyle (A\C)\cup (B\C)\)

In either case, \(\displaystyle (A\cup B)\C\) is a subset of \(\displaystyle (A\C)\cup (B\C)\).

Now, you need to do it the other way: if x is in \(\displaystyle (A\C)\cup (B\C)\) prove that x is in \(\displaystyle (A\ C)\cup (B\C)\).