Proving the Distributive Laws for Sets Using Commutative and Complement Laws

  • Thread starter nicnicman
  • Start date
In summary: Since C' and B' are both complements, we can switch them without changing the result. So it becomes A n (B' n C') which is the same as (A n B') n C'. Is that right?Correct! You've got it. Great job on the summary. In summary, to prove that (A-C) - (B-C) = (A-B) - C, we use the definitions of complement, intersection, and subtraction, as well as DeMorgan's laws and the Commutative and Associative laws. We find that the two expressions are equal by rewriting one in terms of the other and using the Associative laws for intersections. Additionally, we can switch the order of the comple
  • #1
nicnicman
136
0
If A, B, and C are sets prove that (A-C) - (B-C) = (A-B) - C

Note: n = intersection, u = union, and ' = complement.

(A-C)-(B-C)
= (AnC') n (BnC')' by definition of complement, intersection, and subtraction
= (AnC') n (B'uC'') by DeMorgan's laws
= (AnC') n (B'uC) by Complementation law
= A n (C' n (B'uC)) by Commutative laws
= A n ((C'nB') u (C'nC)) by Distributive laws
= A n ((C'nB') u empty set) by Complement laws
= A n (C'nB') by Identity laws
= A n (B'nC') by Commutative laws
= (AnB') n C' by Commutative laws
= (A-B) - C by definition of complement, intersection, and subtraction

How is this? Did I use the Commutative laws correctly?
Thanks for any suggestions.
 
Last edited:
Physics news on Phys.org
  • #2
[itex]\left(A\cap C^c\right)\cap\left(B^c\cup C\right)=A\cap \left(C^c\cap\left(B^c\cup C\right)\right)[/itex] and [itex]A\cap\left(B^c\cap C^c\right)=\left(A\cap B^c\right)\cap C^c[/itex] look more like associative laws than commutative laws to me.

Other than that and the typo (which I'm sure you'll find on your own) it looks good.
 
  • #3
Yeah, those are the parts I was not sure of. I saw similar problems elsewhere and followed their logic, but I'm unsure of how those two steps are using the Associative laws.
Maybe someone could shed some light.

Oh, and I'll fix the typo.

Thanks.
 
  • #4
What does the associative law say for intersections?
 
  • #5
Associative law for intersections:

A n (B n C) = (A n B) n C
 
  • #6
That's correct. Do you see immediately how that applies to [itex]A\cap\left(B^c\cap C^c\right)=\left(A\cap B^c\right)\cap C^c[/itex]?

For [itex](A\cap C^c)\cap(B^c\cup C)=A\cap(C^c\cap(B^c\cup C))[/itex], rewrite that equality with [itex]D[/itex] written in place of [itex](B^c\cup C)[/itex]. Now do you see how the associative law applies here?
 
  • #7
Okay, yeah I see it now. Thanks!

And, the other one is pretty straight forward.
 

FAQ: Proving the Distributive Laws for Sets Using Commutative and Complement Laws

What does it mean to prove a mathematical equation?

To prove a mathematical equation means to show that it is true for all possible values of the variables involved, using logical reasoning and mathematical principles.

Why is proving equations important in science?

Proving equations is important in science because it allows us to validate the accuracy of our theories and predictions, and to establish a solid foundation for further scientific research and discoveries.

What is the purpose of proving (A-C)-(B-C) = (A-B)-C?

The purpose of proving (A-C)-(B-C) = (A-B)-C is to demonstrate the mathematical property of associativity, which states that the grouping of numbers in an equation does not affect the resulting value.

What is the process of proving (A-C)-(B-C) = (A-B)-C?

To prove (A-C)-(B-C) = (A-B)-C, we begin by substituting the values of A, B, and C into the equation. Then, we use algebraic manipulation and logical reasoning to show that both sides of the equation are equal.

What implications does the proof of (A-C)-(B-C) = (A-B)-C have in other branches of science?

The proof of (A-C)-(B-C) = (A-B)-C has implications in various branches of science, as it is a fundamental property of mathematics. It can be applied in fields such as physics, chemistry, and engineering to simplify and solve complex equations, and to validate the accuracy of scientific theories and models.

Similar threads

Prove if ##a\cdot c = b \cdot c## then ##a = b## using Peano postulates
Replies
2
Views
709
Root test proof using Law of Algebra
Replies
6
Views
779
Prove ##(a\cdot b)\cdot c =a\cdot (b \cdot c)## using Peano postulates
Replies
1
Views
727
Prove ##(a+b)\cdot c=a\cdot c+b\cdot c## using Peano postulates
Replies
3
Views
969
Prove ##a\cdot b = b \cdot a ##using Peano postulates
Replies
3
Views
957
Prove ##a + b = b + a## using Peano postulates
Replies
3
Views
1K
Prove ##(a+b) + c = a + (b+c)## using Peano postulates
Replies
9
Views
2K
Solve Set Proof Problems Homework Statement
Replies
2
Views
1K
Understanding Boolean Algebra: Homework Help and Examples
Replies
4
Views
1K
Munkres Topology ch1 ex #9 - Generalized DeMorgan's Laws
Replies
2
Views
1K

Hot Threads

Recent Insights

Back
Top