Verify whether this function is 1-1 and whether it is onto

Question

Question 1 Answer the following questions and show your work:

(a) Let f be a function f : Z → Z x Z such that f(n) = (2n, n + 3). Verify whether this function is 1-1 and whether it is onto.

(b) Let f be a function f : R³ → R such that f(x,y,z) = xyz. Verify whether this function is 1-1 and whether it is onto.

(c) Prove that the function f : R - {2} → R - {5} defined by f(x) = 5x + 1/x-2 is a bijection.

(d) Consider the functions f :R → R,g :R → R x R defined as f (x) = 1/x²+1 and g(x) = (3x,x²). Find the formulas for f o f and g o f.

Question 2 Use truth tables to show the following:

(a) whether ¬p ∧ ¬(p → q) is a tautology, a contradiction or neither.

(b) whether ((p → q) ∧ (p → r)) → (p →) (q ∧ r)) is a tautology, a contradiction or neither.

(c)  ¬(¬p ∧ q) and q → p are logically equivalent.

(d) (p → (q → r)) and (q → (p → r)) are logically equivalent.

Question 3 Use laws of logic (algebraic version) to show the following equivalences (clearly indicate which law you use in each step):

(a) ≡ (q → p).

(b) (p → (q → r)) ≡ (q → (p → r))

(c) ((p → r) V (q → r)) ≡ ((p ∧ q)→ r)

Question 4 Prove the following using the method suggested:

(a) Prove the following either by direct proof or by contraposition:

Let a ∈ Z, if a ≡ 1 (mod 5), then a² ≡ 1 (mod 5).

(6) Prove the following by contradiction:

Suppose a, b ∈ Z. If 4|(a² + b²), then a and b are not both odd.

(c) Disprove the following by counterexamples:

• For every natural number n, the integer n² + 17n + 17 is prime.
• Let A,B and C be sets. If A x C =  B x C, then A= B.

(d) Prove the following by cases: For all n ∈ Z, n² + 3n +4 is even.

(e) Prove the following by induction:

1² + 3² + 5² + ........ + (2n-1)² = n/3 (2n-1) (2n+1)