MHB Proving Set Operations [Set Theory]

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)
 
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  • #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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