In the questions of this problem, we will need you to use the definitions of the various set theory operators in terms of propositional logical identities to prove Set Theory identities. Tables 6, 7, 8 of section 1.3 in your textbook will be helpful for you. To give you an example, for two arbitrary sets A, b U , heres a proof of one of two De Morgans identity laws for sets:

Prove the following set identities using the axioms of set theory, propositional logic, as well as the semantics of the membership operator () as exemplified above.

Problem (2): Proofs of set identities (20 pts) In the questions of this problem, we will need you to use the definitions of the various set theory operators in terms of propositional logical identities to prove Set Theory identities. Tables 6, 7, 8 of section 1.3 in your textbook will be helpful for you. To give you an example, for two arbitrary sets A, b C U, here's a proof of one of two De Morgan's identity laws for sets AUBAnB LHS = AU B Defn. of Universal Complement) (Defn. of (Defn. of union) E(AU B) (De Morgan's law in prop. logic ) ( Defn. 014 ) Defn. of Universal Complement) Defn. of intersection) Prove the following set identities using the axioms of set theory, propositional logic, as well as the semantics of the membership operator (E) as exemplified above. (a) AnB-AUB (b) AU(AnB)-A (c) A (A U B) = A (d) AU(BUC)= (AUB) UC (e) A (B U C) = (A B) U (A C) Symmetric De Morgan's Law fors (Absorption Law for Sets, Version I) Absorption Law for Sets, Version II) Associativity of union) Distributivity, Version I)