# I would like to know for example how I prove or disprove this sentence ?

• MHB
• MathPro17
In summary, the conversation discusses proving the statement "If for any A, B, C sets exist A\C=B\C and A ∩ C = B ∩ C, then A⊆B". It also considers proving the statement "P(A∩B)=P(A)∩P(B)" with the use of set notation. The user is encouraged to continue the series of equivalences to prove the statements.
MathPro17
Hi,
I am new here ,
I have stuck to Proved or disproved This sentence :
If for any A,B,C sets exist A\C=B\C And A ∩ C =B ∩ C , then A⊆ B.

Thanks very much for help.

Last edited:
MathPro17 said:
If for any A,B,C sets exist A\C=B\C And A ∩ C =B ∩ C , then A⊆ B.
Do you mean "it is the case" instead of "exists"? That is, is the claim as follows: If $A\setminus C=B\setminus C$ and $A\cap C=B\cap C$ for some sets $A$, $B$ and $C$, then $A\subseteq B$?

Note that $X\setminus Y=X\cap\overline{Y}$, so $X=X\cap(Y\cup\overline{Y})=(X\cap Y)\cup (X\setminus Y)$. Therefore,
$A=(A\cap C)\cup (A\setminus C)=\text{ (by assumption) }(B\cap C)\cup (B\setminus C)=B.$

Evgeny.Makarov said:
Do you mean "it is the case" instead of "exists"? That is, is the claim as follows: If $A\setminus C=B\setminus C$ and $A\cap C=B\cap C$ for some sets $A$, $B$ and $C$, then $A\subseteq B$?

Note that $X\setminus Y=X\cap\overline{Y}$, so $X=X\cap(Y\cup\overline{Y})=(X\cap Y)\cup (X\setminus Y)$. Therefore,
$A=(A\cap C)\cup (A\setminus C)=\text{ (by assumption) }(B\cap C)\cup (B\setminus C)=B.$

Yes I mean for that .
I would like to your help with another thing - to prove this :
P(A∩B)=P(A)∩P(B)
if you could prove it with:"x ∈ to P()..." ?

Thanks.

MathPro17 said:
I would like to your help with another thing - to prove this :
P(A∩B)=P(A)∩P(B)
if you could prove it with:"x ∈ to P()..." ?
Why don't you continue $X\in P(A\cap B)\iff\dots$ yourself? You'll need $X\subseteq A\cap B\iff X\subseteq A\land X\subseteq B$ where $\land$ means "and".

Evgeny.Makarov said:
Why don't you continue $X\in P(A\cap B)\iff\dots$ yourself? You'll need $X\subseteq A\cap B\iff X\subseteq A\land X\subseteq B$ where $\land$ means "and".

I Mean How could I prove this : "P(A∩B)=P(A)∩P(B)" with Some X that I take ?
Thanks.

MathPro17 said:
I Mean How could I prove this : "P(A∩B)=P(A)∩P(B)" with Some X that I take ?
You need to show $X\in P(A\cap B)\iff X\in P(A)\cap P(B)$ for all sets $X$. Why don't you continue the series of equivalences $X\in P(A\cap B)\iff\dots$ using the definition of powerset?

## 1. How do I prove or disprove a sentence?

In order to prove or disprove a sentence, you must gather evidence and conduct research. This may involve performing experiments, analyzing data, and consulting with other experts in the field. Once you have collected enough evidence, you can make a conclusion about the validity of the sentence.

## 2. What is the scientific method?

The scientific method is a systematic process used by scientists to acquire knowledge and understanding about the natural world. It involves making observations, formulating a hypothesis, conducting experiments, analyzing data, and making conclusions based on evidence.

## 3. How do I determine if a sentence is true or false?

The best way to determine the truth or falsity of a sentence is through the scientific method. By conducting experiments and analyzing data, you can gather evidence to support or refute the sentence. It is important to approach the investigation with an open mind and consider all possible explanations before making a conclusion.

## 4. Can a sentence be proven definitively?

In science, nothing can be proven definitively. The best we can do is gather enough evidence to support a particular explanation or theory. However, as new evidence and technology become available, our understanding and conclusions may change.

## 5. How do scientists handle conflicting evidence?

Conflicting evidence is a common occurrence in science and can be challenging to navigate. Scientists must carefully evaluate and analyze all evidence, considering its reliability and potential biases. They may also consult with other experts in the field and conduct further experiments to gain a better understanding of the issue.

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