Decomposition Based on BCNF is Necessarily Correct

Attributes A, B, C. FD: A C

Relations R1[A,B] R2[A,C]

Tuples in R1: (a,b)

Tuples in R2: (a,c), (a,d)

Tuples in the join of R1 and R2: (a,b,c), (a,b,d)

Can (a,b,d) be a bogus tuple?