Prove or Disprove the following
Let x,y. If x + xy + 1 is even then x is odd
Prove or Disprove the following Let x,y. If x + xy + 1 is even then...
Prove or Disprove #3 (d) For each of the following, prove or disprove: iii) There is an element of X × Y with the form (a, 3a) (d) For each of the following, prove or disprove: iii) There is an element of X × Y with the form (a, 3a)
Prove or Disprove 23. If x <y, then-y -x. Remark 39. What is the analogous problem and answer for bounded below and infima? Prove or Disprove 37. Let SCR be nonempty and bounded below. Then inf(S) exists.
(1) Prove or disprove that if all the elements of a matrix A is even, the determinant of A is even. (2) Compute the following determinant (1) (4 pts) Prove or disprove that if all the elements of a matrix A is even, the determinant of A is even. (2) (2+2 pts) Compute the following determinant (123) (100 A= 1023 B=020 003 co c
1. Prove with a direct proof or disprove by counterexample. If x is an odd integer, then x3 is an odd integer.
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?
Prove or disprove the following: For any (non-directed) graph, the number of odd-degree nodes is even. In a minimally connected graph of n>2 nodes with exactly k nodes of degree 1 , 1<k<n. I.e., you cannot have a minimally connected graph with 1 node of degree 1 or n nodes of degree 1.
prove that there is no solution to xy+yx+xz=1 where x y z are all odd
(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)
Let X, Y E Mn (R). Prove that XY = XY_if and only if there exists an invertible matrix Z so that X = Z In and Y = Z1 + In. Hint: the trace is not involve at all in this problem _
1. Let a, b,cE Z be positive integers. Prove or disprove each of the following (a) If b | c, then gcd(a, b) gcd(a, c). (b) If b c, then ged(a., b) < gcd(a, c)