for the function f gives the equation ________________, which simplifies to ______. This proves that f is one-to-one. Let O be the set of odd integers. Define a function g : O ⟶ ℤ by g(x) = (x + 1)/2. Prove that g is onto. Let n ∈ ℤ be given. Consider the integer j = _________.  First, notice that j ∈ O, since j = 2· ______ + 1. Second, using the formula for g gives g( j) = _______ = n, which proves that g is onto. Let G ⊆ ? (ℕ) be the set of all subsets of natural numbers. Define a function m : G ⟶ ℕ by taking m(S) to be the smallest number in the set S. Give a counterexample to show that m is not one-to-one.

1. Let f : ℕ ⟶ ℕ be defined by f(x) = 10x + 5. Fill in the blanks in the following proof that f is one-to-one.
   Let a, b ∈ ℕ and suppose that _______________. Using the formula for the function f gives the equation ________________, which simplifies to ______. This proves that f is one-to-one.
2. Let O be the set of odd integers. Define a function g : O ⟶ ℤ by g(x) = (x + 1)/2. Prove that g is onto.
   Let n ∈ ℤ be given. Consider the integer j = _________.  First, notice that j ∈ O, since j = 2· ______ + 1. Second, using the formula for g gives g( j) = _______ = n, which proves that g is onto.
3. Let G ⊆ ? (ℕ) be the set of all subsets of natural numbers. Define a function
   m : G ⟶ ℕ by taking m(S) to be the smallest number in the set S. Give a counterexample to show that m is not one-to-one. 
4. Let T be the set of all triangles, and define A : T ⟶ R by taking A(t) to be the degree measure of the largest angle of the triangle t. Explain why A is not onto.

Define a relation R on Z by a R b if a = b + 5k for some integer k.

1. Complete the following proof that R is reflexive.
   Let a ∈ ______. Since a = a + 5·______, a R a, as required.
2. Complete the following proof that R is symmetric.
   Let a, b ∈ Z and suppose that a R b. By the definition of R, a = _____ for some integer k. Using algebra, b = ________, which shows that ______, as required.
3. Complete the following proof that R is transitive.
   Let a, b, c ∈ Z and suppose that ______ and ______. By the definition of R, a = ______ for some integer k1 and b = ______ for some integer k2. Substituting the second equation into the first gives a = _________________ = c + 5(_____),  which shows that _______, as required.