Abstract Algebra Theory and Applications - Computer Science ...

EXERCISES 197. Prove A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C).8. Prove A ⊂ B if and only if A ∩ B = A.9. Prove (A ∩ B) ′ = A ′ ∪ B ′ .10. Prove A ∪ B = (A ∩ B) ∪ (A \ B) ∪ (B \ A).11. Prove (A ∪ B) × C = (A × C) ∪ (B × C).12. Prove (A ∩ B) \ B = ∅.13. Prove (A ∪ B) \ B = A \ B.14. Prove A \ (B ∪ C) = (A \ B) ∩ (A \ C).15. Prove A ∩ (B \ C) = (A ∩ B) \ (A ∩ C).16. Prove (A \ B) ∪ (B \ C) = (A ∪ B) \ (A ∩ B).17. Which of the following relations f : Q → Q define a mapping? In each case,supply a reason why f is or is not a mapping.(a) f(p/q) = p + 1p − 2(b) f(p/q) = 3p3q(c) f(p/q) = p + qq 2(d) f(p/q) = 3p27q 2 − p q18. Determine which of the following functions are one-to-one and which areonto. If the function is not onto, determine its range.(a) f : R → R defined by f(x) = e x(b) f : Z → Z defined by f(n) = n 2 + 3(c) f : R → R defined by f(x) = sin x(d) f : Z → Z defined by f(x) = x 219. Let f : A → B and g : B → C be invertible mappings; that is, mappingssuch that f −1 and g −1 exist. Show that (g ◦ f) −1 = f −1 ◦ g −1 .20. (a) Define a function f : N → N that is one-to-one but not onto.(b) Define a function f : N → N that is onto but not one-to-one.21. Prove the relation defined on R 2 by (x 1 , y 1 ) ∼ (x 2 , y 2 ) if x 2 1 + y 2 1 = x 2 2 + y 2 2 isan equivalence relation.22. Let f : A → B and g : B → C be maps.(a) If f and g are both one-to-one functions, show that g ◦ f is one-to-one.(b) If g ◦ f is onto, show that g is onto.(c) If g ◦ f is one-to-one, show that f is one-to-one.(d) If g ◦ f is one-to-one and f is onto, show that g is one-to-one.