# This would be a false statement, correct?

• I

## 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? Related Set Theory, Logic, Probability, Statistics News on Phys.org
BvU
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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

• Mark44 and Dale
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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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 $A \cap B$.

• FactChecker and pbuk
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
I believe that I am correct, the following statement here must be FALSE, right?
To reinforce what others have said, there is a difference between talking about a "statement" versus taking about a "propositional function".

If you utter a phrase like "If x > 7 then x > 1" , it is technically a "propositional function", which cannot be assigned a truth value ( True vs False) until "x" is defined to be something specific. In common speech and in informal mathematical writing, it is usually understood that phrases like "If x >7 then x > 1" are intended to have the "universal quantifier" given by the phrase "for each". So the reader interprets the claim "If x > 7 then x > 1" to mean "For each number x, if x >7 then x > 1". With this interpretation, the phrase becomes a statement which can be assigned a truth value.

The other commonly used quantifier is "there exists". Someone writing hasty notes might jot down the phrase "x > 1" intending it to mean "There exists a number x such that x > 1". However, this is not a clear style of writing.

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.

• • PeroK and pbuk