Assignment Question 1 Two sets are identical if they have the same elements In our case, we need a model of sets in which the extensionality axiom does not hold i.e. a≠ b It follows that: U= Where a ϵ {a}, a ϵ {a, b} and because a≠ b; then {a} ≠ {a, b} but for all x ϵ U, we have X ϵ {a} ↔ x, ϵ {a, b} Using the above relation, we thus have sets b & c and sets a & d as identical. Question 2 Part 1 To prove that = for a= c and b= d We first define axiom extensionality Ɐx Ɐy (x =y ↔ Ɐz (z ϵ x↔ z ϵ y) Assuming all x=a and y=b Then; A ϵ {a} and a ϵ {a, b} Now that a=b, we have {a}= {a, b} But for every x ϵ U We get x ϵ {a} ↔ x ϵ {a,...
Words: 275
Pages: 1