Question

R₁ = {(4,5)}

R₂ =    {(1,5), (1,6), (1.7), (1,8), (2,5), (2,6), (2,7), (2,8), (3,5), (3,6), (3,7), (3,8), (4,5), (4,6), (4,7), (4,8)}

Evaluate R₁ ◦ R₂ and R₂ ◦ R₁

Answer 1

Recall that $(x,z)\in A\circ B$ if and only if there exists $y$ such that $(x,y)\in A$ and $(y,z)\in B.$

Since there is no pair $(5,y)$ in the relation $R_2,$ we conclude that $R_1\circ R_2=\emptyset.$ By analogy, since there is no pair $(x,4)$ in the relation $R_2,$ we conclude that $R_2\circ R_1=\emptyset.$