366staxioms

# 366staxioms - A and B A ⊆ B iﬀ ∀ x x ∈ A → x ∈...

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SET THEORY AXIOMS Math 366 1. (Extensionality Axiom) For any sets A and B , if x ( x A iﬀ x B ) then A = B . 2. (Deﬁnition of the Emptyset) (a) is a set. (b) For all x , x 6∈ ∅ . 3. (Deﬁnition of Intersection) (a) For all sets A and B , A B is a set. (b) For all sets A and B , for all x , x A B iﬀ x A and x B 4. (Deﬁnition of Union) (a) For all sets A and B , A B is a set. (b) For all sets A and B , for all x , x A B iﬀ x A or x B 5. (Deﬁnition of Relative Complement) (a) For all sets A and B , A - B is a set. (b) For all sets A and B , for all x , x A - B iﬀ x A and x 6∈ B 6. (Deﬁnition of ) For all sets

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Unformatted text preview: A and B , A ⊆ B iﬀ ∀ x ( x ∈ A → x ∈ B ) 7. (Deﬁnition of Power Set) (a) For any set A , P ( A ) is a set. 1 (b) For any set A , for all x , x ∈ P ( A ) iﬀ x ⊆ A 8. (Pairing Axiom) For all x , y , u and v , ( x, y ) = ( u, v ) iﬀ x = y and u = v 9. (Deﬁnition of Cartesian Products) (a) For all sets A and B , A × B is a set. (b) For all sets A and B , for all x , x ∈ A × B iﬀ ∃ a ∈ A ∃ b ∈ B s.t. x = ( a, b ) 2...
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366staxioms - A and B A ⊆ B iﬀ ∀ x x ∈ A → x ∈...

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