For any sets A, B and C, let R ₁ be a relation from A to B and R ₂ be a _relation from B to C, Define the composition of R ₁ and R ₂ to be the relation from A to C given by

(b) Prove that for any sets A, B, C, D and for any relations R ₁ from A to B, R ₂ from B to C and R ₃ from C to D, we have

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For any sets A, B and C, let R₁ be a relation from A to B and R₂ be a relation from B to C. Define the composition of R₁ and R₂ to be the relation from A to C given by R₂ R₁ = {(a, c) AxC: (a,b) = R₁ and (b, c) € R₂ for some b € B} CAX C. (a) Let A = {1,2,3,4}, B = {5,6,7} and C = {8,9,10). Consider the relations R₁ = {(1,5), (1,7), (2,6), (2,7), (3,5), (4, 5), (4,6), (4,7)} from A to B and R₂ = {(5, 10), (6, 8), (6,9), (6, 10), (7,9), (7,10)} from B to C. Write down the composition R₂ o R₁ from A to C. R₂0 (R₂0 R₁) (R₂0 R₂)0 R₁. For any sets A, B and C, let R₁ be a relation from A to B and R₂ be a relation from B to C. Define the composition of R₁ and R₂ to be the relation from A to C given by R₂ R₁ = {(a, c) AxC: (a,b) = R₁ and (b, c) € R₂ for some b € B} CAX C. (a) Let A = {1,2,3,4}, B = {5,6,7} and C = {8,9,10). Consider the relations R₁ = {(1,5), (1,7), (2,6), (2,7), (3,5), (4, 5), (4,6), (4,7)} from A to B and R₂ = {(5, 10), (6, 8), (6,9), (6, 10), (7,9), (7,10)} from B to C. Write down the composition R₂ o R₁ from A to C. R₂0 (R₂0 R₁) (R₂0 R₂)0 R₁.