1. (20pts) Prove or disprove each of the following statements. If true, then write a proof...
disprove the following statements (if it is true, please write a proof 1: (15 Points) Prove or or quote the corresponding theorem from the textbook; if it is false, please provide a counter example to disprove If u is orthogonal to all the vectors 1, U2,,n then u is orthogonal to all the vectors in Span({, ,., )
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....
Of the following statements, one is true and one is false. Prove the true statement, and for the false statement, write out its negation and prove that. (a) For all sets A, B and C, if(ANB) - C = Ø, then (AUB) CC. (b) , For all sets A, B and C, if (AUB) CC, then (An:B) - C = Ø.
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) Prove or disprove the following statements. (a) Let a, b and c be integers. If aſc and b|c, then (a + b)|c (b) Let a, b and c be integers. If aſb, then (ac)(bc)
(3 + 3 = 6 pts.) Prove or disprove the following statements. If you are proving a statement, then give proper reasoning. If you are disproving a statement, then it is enough to give an example which demonstrates that the statement is false. i. If A and B are two n x n matrices, then (A + B)2 = A + 2AB + B2. ii. Let A be a nxn matrix and let I be the n x n identity...
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)
The following statement is either true or false. If the statement is true, prove it. If the statement is false, give a specific counterexample... If A, B, C and D are sets, then (A × B)∩(C × D) = (A ∩ C)×(B ∩ D).
4. True or False. If the statement is true, give a proof. If it is false, give one example showing it is false. a, b, c are integers. (1) a) If a|c and b|c then ab|c. (1) b) If a|bc then a|b or a|c. (1) c) If a|(b^2)then a 2|(b^4) .
help please and thank you 2. Prove that the following statements are true for sets A, B, C: (a) Commutativity (I): An B = BNA. (b) Commutativity (II): AU B = BU A. (c) Distributivity (I): AN(BUC) = (AN B)U(ANC). (d) Distributivity (II): AU (BAC) = (AUB) N (AUC). (e) Idempotence (I): An A = A. (f) Idempotence (II): AU A = A.