2. [5 points] Consider the orthogonality relation R on the set B = {(0,0,0), (1,0,0), (0,1,0),...

    

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2. [5 points] Consider the orthogonality relation R on the set B = {(0,0,0), (1,0,0), (0,1,0), (1,1,0), (0,0,1), (1,0,1), (0,1,1), (1,1,1)), given by (X1, X2, X3) R (Y1, V2, Y3) (X1'Y + X2'Y2 + X3'Y3) mod 2 = 0, for all (X1, X2, X3), (1, V2, V3) E B. (a) Argue that R is symmetric. (b) Give an example showing that R is not transitive, i.e., find some (X1, X2, X3), (Y1, V2, Y3), and (Z1, Z2, Z3) such that (X1, X2, X3) is orthogonal to (Y1, V2, V3) and (Y1, V2, Y3) is orthogonal to (Z1, Z2, Z3), but (X1, X2, X3) is not orthogonal to (Z1, Z2, Z3). 2. [5 points] Consider the orthogonality relation R on the set B = {(0,0,0), (1,0,0), (0,1,0), (1,1,0), (0,0,1), (1,0,1), (0,1,1), (1,1,1)), given by (X1, X2, X3) R (Y1, V2, Y3) (X1'Y + X2'Y2 + X3'Y3) mod 2 = 0, for all (X1, X2, X3), (1, V2, V3) E B. (a) Argue that R is symmetric. (b) Give an example showing that R is not transitive, i.e., find some (X1, X2, X3), (Y1, V2, Y3), and (Z1, Z2, Z3) such that (X1, X2, X3) is orthogonal to (Y1, V2, V3) and (Y1, V2, Y3) is orthogonal to (Z1, Z2, Z3), but (X1, X2, X3) is not orthogonal to (Z1, Z2, Z3).