A Crash Course in Set Theory

Definition. A specific set that we often care about is the empty set, ∅ i.e. the set containing no elements. One particular quirk of the empty set is that any statement of the form ∀x ∈ ∅ . . . will always be vacuously true, as it is impossible to disprove (as we disprove ∀ claims by using ∃ quantifiers!) For example, the statement “every element of the empty set is delicious” is true. Dumb, but true! Some other frequently-occurring sets are the open intervals (a, b) = {x | x ∈ R, a < x < b} and closed intervals [a, b] = {x | x ∈ R, a ≤ x ≤ b}.