disprove the following statements (if it is true, please write a proof 1: (15 Points) Prove...
1. (20pts) Prove or disprove each of the following statements. If true, then write a proof for the statement. If false, then give a specific explicit example. a) {12a + 4b: a and b are integers} = {4c: c is an integer), and b) For sets A, B and C: A(BUC)=(A\B)U(A\C).
Problem 1. Determine whether the following statements are True or False, and provide a short proof (or a counter-example) of your claim. (a) If A is an orthogonal matrix then A² is orthogonal. (b) If A2 is an orthogonal matrix then A is orthogonal.
Write a formal proof to prove the following conjecture to be
true or false.
If the statement is true, write a formal proof of it. If the
statement is false, provide a counterexample and a slightly
modified statement that is true and write a formal proof of your
new statement.
Conjecture:
15. (12 pts) Let h: R + RxR be the function given by h(x) = (x²,6x + 1) (a) Determine if h is an injection. If yes, prove it....
3. Prove the statements that are true and give counterexamples to disprove those that are false. (a). Va,b,n E Z* , if a’ =b}(modn) then a =b(modn). (8 points) (b). If p> 2 and q> 2 are prime, then p? +q must be composite. (12 points)
(1 point) Are the following statements true or false? ? 1. If z is orthogonal to uị and u2 span(uj, u2), then z must be in and if W = Wt. ? 2. For each y and each subspace W, the vector y – projw(y) is orthogonal to W. ? 3. If y is in a subspace W, then the orthogonal projection of y onto W is y itself. ? 4. The orthogonal projection p of y onto a subspace...
Problem 1. (15 points) Answer the following true or false (ao proof or argurment needed). (a). True or False: solutions. There exists a system of linear equations which has exactly two TrUR (b). True or False: most one IfA is an m x n matrix with null(A) = 0 then AE = 6 has at solution. yhjL (c). True or False: If A and B are invertible nxn matrices then AB is invertible and (AB)-1 = A-B- Fals R. Then...
4. (15 points) Prove or disprove each one of the following statements: ·0(f(n) + g(n)) = f(n) + 0(g(n))I f(n) and g(n) are strictly positive for all n ·0(f(n) × g(n)) positive for all n f(n) 0(g(n))I f(n) and g(n) are strictly
Let X, Y, Z be random variables. Prove or disprove the following statements. (That means, you need to either write down a formal proof, or give a counterexample.) (a) If X and Y are (unconditionally) independent, is it true that X and Y are conditionally indepen- dent given Z? (b) If X and Y are conditionally independent given Z, is it true that X and Y are (unconditionally) independent?
(1 point) Are the following statements true or false? ? 1. u? v – vſ u = 0. ? 2. If x is orthogonal to every vector in a subspace W , then x is in Wt. ? 3. For any scalar c, ||cv|| = c||v. ? 4. For an m x n matrix A, vectors in the null space of A are orthogonal to vectors in the row space of A. ? 5. If u and v are nonzero...
Detailed proof please.
. 1. Determine whether the following statements are true or false. If one is true, provide a proof. If one is false, provide a counterexample (proving that it is in fact a counterexample). IF f is a positive continuous function on [1,00) and (f(x))2dx converges, THEN Sº f(x)dx converges. • IF f is a positive continuous function on [1,00) such that limx700 f(x) O and soon f(x)dx converges, THEN S ° (f (x))2dx converges. IF f is...