Prove the following statements using algebraic manipulation.a) (15 pts) If s = x⊕y⊕c, please prove c = x⊕y⊕s.b) (15 pts)b-1): 0⊕x = x;b-2): 1⊕x = xb-3): x⊕x = 1c) (15 pts) xyz + xyz + xyz + xyz = xy + yz + xzd) (15 pts) Prove that the XOR operation is associate, that is: x⊕(y⊕z) = (x⊕y)⊕z2) (20 pts) Obtain the minimized logic function in the form of product-of-sums (SOP) for the followingincompletely specified truth table, through use of a Karnaugh map.Row # x y z w f0 0 0 0 0 d1 0 0 0 1 d2 0 0 1 0 13 0 0 1 1 04 0 1 0 0 05 0 1 0 1 16 0 1 1 0 07 0 1 1 1 18 1 0 0 0 d9 1 0 0 1 010 1 0 1 0 111 1 0 1 1 012 1 1 0 0 113 1 1 0 1 114 1 1 1 0 015 1 1 1 1 d