# This would be a false statement, correct?

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## Summary:

A quick question about sets and intersection/unions

## Main Question or Discussion Point

I believe that I am correct, the following statement here must be FALSE, right? It has to be false because A union B is like the two entire circles of the Venn diagram and that cannot be a subset of the intersection area, right? Now if this statement was flipped, then it would be true?

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BvU
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2019 Award
Correct !

BvU
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2019 Award
By the way PF would really appreciate typed posts instead of disk-space wasting big pictures !

If you type
$A\cup B \subseteq A\cap B$ then you get $A\cup B \subseteq A\cap B$ (in-line math), and if you type
$$A\cup B \subseteq A\cap B$$ then you get $$A\cup B \subseteq A\cap B$$
('displayed math').

$LaTeX$ is fun, fairly easy to start with and extremely useful

Math_QED
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Yes correct. For example take $A = \{0\}, B = \{1\}$. Then $A\cup B = \{0,1\}$ yet $A \cap B = \emptyset$.

The other inclusion, that is $A \cap B \subseteq A \cup B$, is trivially true.

PeroK
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Gold Member
Summary:: A quick question about sets and intersection/unions

I believe that I am correct, the following statement here must be FALSE, right? It has to be false because A union B is like the two entire circles of the Venn diagram and that cannot be a subset of the intersection area, right? Now if this statement was flipped, then it would be true?
You can, however, have $A \cup B = A \cap B$, so it's not always false. Can you see when you have this equality?

It is correct that $A \cup B$ can never be a proper subset of $A \cap B$.

pbuk
Gold Member
You can, however, have $A \cup B = A \cap B$, so it's not always false. Can you see when you have this equality?
Yes, absolutely, it is NOT correct to say that the statement is unconditionally false.

Stephen Tashi
Your question about "$A \cup B \subseteq A \cap B$" is technically a question about a propositional function, so it cannot be assigned a single truth value. You probably intended some universal quantifiers to be supplied, so the phrase would become the statement "For each set $A$ and for each set $B$, $A \cup B \subseteq A \cap B$". With the universal quantifiers, the propositional function is converted into a False statement. However, if we supply existential quantifiers, we get "There exists a set $A$ and there exists a set $B$ such that $A \cup B \subseteq A \cap B$". This converts the propositional function into a True statement.