Writing Assignment-2 Prove the following by using-a direct argument

Writing Assignment #2#1) Prove the following by using: a direct argument; the definition for union; thedefinition for intersection; and the definition for set complement.Let A , B , and U be sets. If U= A ? B and #2) Prove the following by using: a direct argument; and definition 2.1.16.Consider practice problem 2.1.14 as an outline for your proof.Let B { A j s . t . j ?J } be an indexed family of sets and let be a set. B ? ( ¿ j? J A j ) =¿ j ? J ( B ? A j )#3) Prove the following by using: a direct argument; definition 2.2.9; definition of arational number; sum and product of integers are integers.Let P y , i.e. be a relation on x P y , if x? y R defined as: ? x , y? R , x is related to is rational. #4) Prove the following by using: a contradiction argument; definition 2.3.1;definition and notation 2.3.13; and Theorem 2.3.16.Let f : A?B , S ? A , and T?A . S ?T