Proving Set Operations [Set Theory]

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Discussion Overview

The discussion revolves around proving or disproving a set operation equation involving three sets A, B, and C. Participants explore various theorems and rules related to set operations, including the commutative property, distributive property, and DeMorgan's laws. The focus is on manipulating expressions involving set differences and intersections.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant requests assistance in proving or disproving the equation A−(B∩C) = (A−B)∩(A−C) using set operation theorems.
  • Another participant suggests replacing set differences with intersections and complements to simplify the expression.
  • Some participants express uncertainty about using the distributive property due to the presence of both union and intersection operations.
  • A participant proposes that if A=B and C is non-empty, a counterexample could demonstrate that the equality does not hold.
  • There is a discussion about the necessity of showing effort in solving the problem before receiving full solutions from others.
  • One participant mentions that they have derived an expression using distributive properties but is unsure of the next steps.
  • Another participant emphasizes the importance of exploring counterexamples to disprove the statement.

Areas of Agreement / Disagreement

Participants express differing views on how to approach the proof or disproof of the equation. Some suggest using counterexamples, while others focus on manipulating the expressions using set operation rules. There is no consensus on the correct method or outcome of the discussion.

Contextual Notes

Some participants indicate they are still learning the theorems related to set operations, which may limit their ability to engage fully with the problem. The discussion includes various assumptions about the properties of sets and the operations being performed.

Who May Find This Useful

This discussion may be useful for students studying set theory, particularly those interested in understanding the application of set operation theorems and exploring proofs or counterexamples in mathematical reasoning.

TheGreat
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Could someone help me in this by simplifying/Proving the equation using Theorems/ Rules on Operation of Sets (i.e Commutative property, idempotent, assoc, dist. , definition of union , def. intersection , DeMorgan ,etc.).

Letting A, B and C be three sets ..

Prove/Disprove :
View attachment 5616
Any solution would be much of help. Hoping to get answers .. Do help me :) Thank you!

I've seen similar questions in here like this:
Problem posted: Prove/Disprove : A−(B∩C) = (A−B)∩(A−C)
soroban said:
Hello, KOO!

We should work on one side of the equation.


\begin{array}{cccccc}<br /> 1. &amp; A -(B \cap C) &amp;&amp; 1. &amp;\text{Given} \\<br /> 2. &amp; A \cap(B\cup C)^c &amp;&amp; 2. &amp;\text{def. Subtr&#039;n} \\<br /> 3. &amp; A \cap B^c \cap C^c &amp;&amp; 3. &amp; \text{DeMorgan} \\<br /> 4. &amp; A \cap A \cap B^c \cap C^c &amp;&amp; 4. &amp; \text{Duplication} \\<br /> 5. &amp; A\cap B^c \cap A \cap C^c &amp;&amp; 5. &amp; \text{Commutative} \\<br /> 6. &amp; (A \cap B^c) \cap (A \cap C^c) &amp;&amp; 6. &amp; \text{Associative} \\<br /> 7. &amp; (A-B) \cap (A-C) &amp;&amp; 7. &amp; \text{def. Subtr&#039;n}\end{array}

Thank you once again ... Hoping I could get some answer also ..

- jer
 

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TheGreat said:
Could someone help me in this by simplifying/Proving the equation using Theorems/ Rules on Operation of Sets (i.e Commutative property, idempotent, assoc, dist. , definition of union , def. intersection , DeMorgan ,etc.).

Letting A, B and C be three sets ..

Prove/Disprove :
https://www.physicsforums.com/attachments/5616
Any solution would be much of help. Hoping to get answers .. Do help me :) Thank you!

Greetings TheGreat! Welcome to MHB! (Wave)

Let's start by replacing the subtractions by regular set operations.
That is, replace $A-B$ by $A \cap B^c$, and replace $C^c - A^c$ by $C^c \cap (A^c)^c = C^c \cap A$.
It makes it a bit easier to see where the left hand expression should go.
We'll get:
$$ (A^c \cup C^c) \cap (A\cap B^c) \overset?= C^c \cap A$$

Next, let's apply distributivity to the left hand side.
That is, apply the rule that $(P\cup Q)\cap R$ is the same as $(P \cap R) \cup (Q \cap R)$.
Can you? (Wondering)
 
I am still a student and I have not yet memorized all theorems on set operations .. I will try to read more on theorems .. Anyway, thank you very much ..
If ever you could prove/disprove it with the full steps, it be very much of a big help :) thanks
 
TheGreat said:
I am still a student and I have not yet memorized all theorems on set operations .. I will try to read more on theorems .. Anyway, thank you very much ..
If ever you could prove/disprove it with the full steps, it be very much of a big help :) thanks

Sorry, but we don't usually give full solutions... we believe that is counter productive.
That's why I tried to give as many hints as possible, but we believe a little effort from the poster is required to achieve anything.
If you're not up to showing any effort, or giving any indication where you're stuck, I don't see how we can "really" help you. (Shake)
 
ah okay sorry, I am trying to solve it now . Anyway on the LHS, I can't seem to use distributive because operations used inside the parenthesis are union and intersection. How's that?

I like Serena said:
$$ (A^c \cup C^c) \cap (A\cap B^c) \overset?= C^c \cap A$$

Next, let's apply distributivity to the left hand side.
That is, apply the rule that $(P\cup Q)\cap R$ is the same as $(P \cap R) \cup (Q \cap R)$.
 
TheGreat said:
ah okay sorry, I am trying to solve it now . Anyway on the LHS, I can't seem to use distributive because operations used inside the parenthesis are union and intersection. How's that?

I like Serena said:
Greetings TheGreat! Welcome to MHB! (Wave)

Let's start by replacing the subtractions by regular set operations.
That is, replace $A-B$ by $A \cap B^c$, and replace $C^c - A^c$ by $C^c \cap (A^c)^c = C^c \cap A$.
It makes it a bit easier to see where the left hand expression should go.
We'll get:
$$ (A^c \cup C^c) \cap (A\cap B^c) \overset?= C^c \cap A$$

Next, let's apply distributivity to the left hand side.
That is, apply the rule that $(P\cup Q)\cap R$ is the same as $(P \cap R) \cup (Q \cap R)$.
Can you? (Wondering)

It means that we pick $P=A^c$, $Q=C^c$, and $R=(A\cap B^c)$.
If we do that, what would $(P \cap R) \cup (Q \cap R)$ look like? (Wondering)
 
ah okay , tried distributive and I came up with:
[$A^c$ ∩ (A ∩ $B^c$ )] ∪ [$C^c$ ∩ ( A ∩ $B^c$ )]

what should I do next. Should I use Associative Property to eliminate $A^c$ ∩ A and come up with θ ?
 
TheGreat said:
ah okay , tried distributive and I came up with:
[$A^c$ ∩ (A ∩ $B^c$ )] ∪ [$C^c$ ∩ ( A ∩ $B^c$ )]

what should I do next. Should I use Associative Property to eliminate $A^c$ ∩ A and come up with θ ?

Yep! Keep going! (Nod)
 
TheGreat said:
Could someone help me in this by simplifying/Proving the equation using Theorems/ Rules on Operation of Sets (i.e Commutative property, idempotent, assoc, dist. , definition of union , def. intersection , DeMorgan ,etc.).

Letting A, B and C be three sets ..

Prove/Disprove :

Any solution would be much of help. Hoping to get answers .. Do help me :) Thank you!

I've seen similar questions in here like this:
Problem posted: Prove/Disprove : A−(B∩C) = (A−B)∩(A−C)Thank you once again ... Hoping I could get some answer also ..

- jer

1st of all if the problem is prove/disprove andvnot only prove ,you should 1st try to disprove the problem, by a counter example because that will save you a lot of unnecessary work.

2ndly the problem that you recalled should read :A−(BUC) = (A−B)∩(A−C) and not

A−(B∩C) = (A−B)∩(A−C)
 
Last edited:
  • #10
I like Serena said:
Yep! Keep going! (Nod)

And how are you going to prove that:

$$A\cap B^c\cap C^c$$ it is equal to $$A\cap C^c$$ ?
 
  • #11
I like Serena said:
Greetings TheGreat! Welcome to MHB! (Wave)

Let's start by replacing the subtractions by regular set operations.
That is, replace $A-B$ by $A \cap B^c$, and replace $C^c - A^c$ by $C^c \cap (A^c)^c = C^c \cap A$.
It makes it a bit easier to see where the left hand expression should go.
We'll get:
$$ (A^c \cup C^c) \cap (A\cap B^c) \overset?= C^c \cap A$$

Next, let's apply distributivity to the left hand side.
That is, apply the rule that $(P\cup Q)\cap R$ is the same as $(P \cap R) \cup (Q \cap R)$.
Can you? (Wondering)
COUNTER EXAMPLE

If in the above identity of sets we put A=B ,where A and B are non empty and let $$C^c$$ be non empty,​then the above equality is not satisfied
 
  • #12
solakis said:
And how are you going to prove that:

$$A\cap B^c\cap C^c$$ it is equal to $$A\cap C^c$$ ?

Heh. When we have that, we have exactly what we need to set up a counter example to disprove it.
 

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