Set-Theory Flashcards
What is the intersection between F and G?
The intersection between F and G is the set of all things which are both F and G.
What is the union of F and G?
The union of F and G is the set containing F and G.
What does it mean for two sets to be disjoint?
Two sets are disjoint when they do not share any elements.
What is the difference between a subset and a member?
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Consider (∀x)(Fx ⊃ Gx).
How do we say it in ordinary English? What interpretations make it true?
All Fs are Gs.
The extension of F is a subset of the extension of G.
Consider (∃x)(Fx . Gx).
How do we say it in ordinary English? What interpretations make it true?
Some Fs are Gs.
The intersection of F and G is not the empty set.
Consider (∀x)(Fx . Gx).
How do we say it in ordinary English? What interpretations make it true?
Everything is both F and G.
The intersection of F and G is the UD.
Consider (∀x)(Fx v Gx).
How do we say it in ordinary English? What interpretations make it true?
Everything is either an F or a G.
The union of F and G is the UD.
Consider (∃x)(Fx v Gx).
How do we say it in ordinary English? What interpretations make it true?
Something is either an F or a G.
The union of F and G is not the empty set.
Consider (∀x)(Fx ≡ Gx).
How do we say it in ordinary English? What interpretations make it true?
All and only Fs are Gs.
The extensions of F and G are the same.
Consider (∃x)(Fx ≡ Gx).
How do we say it in ordinary English? What interpretations make it true?
No English variant.
Atleast one thing is either in both F and G or neither F or G.
(biconditionals are conjunctions of conditionals. There are no ordinary English variants for existentially quantified conditionals.(
Consider (∃x)(Fx ⊃ Gx).
How do we say it in ordinary English? What interpretations make it true?
No English variant.
Atleast one thing is either not in F or is in G.
(there are no English variants for existentially quantified conditionals.)
Consider -(∃x)(Fx . Gx).
How do we say it in ordinary English? What interpretations make it true?
No F is G.
The extensions of F and G are disjoint.
Consider (∃x)(Fx . -Gx).
How do we say it in ordinary English? What interpretations make it true?
Some F is not G.
There is something which is in F but not in G.
Consider (∀x)(Fx ⊃ -Gx).
How do we say it in ordinary English? What interpretations make it true?
No F is G.
The extensions of F and G are disjoint.
