1. Prove the following statement by using a contrapositive. "For all x E R, if cos(x)...
yl -y Using the fact that (cos x)' = - sin x, prove that | cos x- cos for all , yE R. (Hint: MVT)
8. a. Prove that f(x) = cos x is continuous on R. b. If ECR and f: E R is continuous on E, prove that 8(x) = cos (f(x)) is continuous on E. 9. For each of the following equations, determine the largest subset E of R such that the given equation
prove the following by contrapositive: If n | ((n − 1)! + 1), then n is prime.
(6) Use a proof by contrapositive to prove for all integers a, b and c, if a t be then à f 6. (7) Prove using cases that the square of any integer has the form 4k or 4k +1 for some integer k. (8) Prove by induction that 32n -1 is divisible by 8.
Suppose that (a-r, a) C E or (a, a + r) C E, f : E → R, L E R, and (1) Prove that there exist numbers 0 < δ < r and M > 0 such that If(x)| < M for all (2) Prove that if L is nonzero, then there exist numbers 0 < δ < r and η > 0 such that limx→af(x) = L xEEwith 0 < |x-a| < δ. If(x)| > η for all...
Consider the statement: "Let r, y e Z. If ry is even, then r is even or y is even.” (1) Write down the converse of the implication. (2) Write down the contrapositive of the implication. (3) Prove the statement by contrapositve.
Discrete Structures
Name: Problem 2. Prove the following theorem using P Theorem. Let x, y e Z. If c-y is odd, then 1 em using proof by contrapositive. yis odd, then ris odd or y is odd.
4. Prove the following statement: Consider the ODE x = f(x) with x : J C R → Rn and f : Rn → Rn. If a continuously differentiable real-valued function V = V(x) exists such that (a) V is defined on Bs(0) {x E Rn : Irl < δ} (b) V(x) 0 for x E Bs(0) 1 fo) (c) V 0) 1 (o then the origin is unstable. (x) >0 for rE Bs
4. Prove the following statement: Consider...
a) Let f : R → R be a function and CER. Definition 1. The lim+oe an A if for every e>0 there erists a M EN such that for all n 2 M we have lan - A<E Complete the following statement with out using negative words (you do not have to prove it): The lim, 10 10if R).Consider the following subsets of P: (b) Let P2-(f(t)- ao at + azt | ao, a1, a2 and Notice that Y...
5. Prove the following identity. (sin x + cos x) = 1+sin 2x 6. If cos(x) = , and x E QIV , find the exact value of each of the following. a.sin (2x) b.cos cos()