1. Let (T,A,V,0,1) be a Boolean Algebra. Define TxTT and o: Tx TT as follows: x*y:=...

 

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1. Let (T,A,V,0,1) be a Boolean Algebra. Define TxTT and o: Tx TT as follows: x*y:= (x Vy)' roy := (x^y)' (a) Show, using the laws of Boolean Algebra, how to define ry using only x, y, and parentheses. (b) Show, using the laws of Boolean Algebra, how to define oy using only x, y, and parentheses. Define RCT T as follows: (x,y) ER if, and only if, (Ay) V (Ay)=1 (10 marks) (c) Show, using the laws of Boolean Algebra, that R is an equivalence relation. Hint: You may want to use the observation that if A=B=1 then AABAC AAB implies C-1 (why?) (10 marks) 1. Let (T,A,V,0,1) be a Boolean Algebra. Define TxTT and o: Tx TT as follows: x*y:= (x Vy)' roy := (x^y)' (a) Show, using the laws of Boolean Algebra, how to define ry using only x, y, and parentheses. (b) Show, using the laws of Boolean Algebra, how to define oy using only x, y, and parentheses. Define RCT T as follows: (x,y) ER if, and only if, (Ay) V (Ay)=1 (10 marks) (c) Show, using the laws of Boolean Algebra, that R is an equivalence relation. Hint: You may want to use the observation that if A=B=1 then AABAC AAB implies C-1 (why?) (10 marks)