Prove or disprove the following statements: a) If both x2 and x3 are rational, then so...

Prove or disprove the following statements:

a) If both x² and x³ are rational, then so is x.

b) If both x² and x³ are irrational, then so is x.

c) If both x+y and xy are rational, then so are x and y.

Expert Solution

Colby Messinger answered 1 year ago

Prove or disprove the following statements. (a) There is a simple graph with 6 vertices with...

Prove or disprove the following statements. (a) There is a simple graph with 6 vertices with degree sequence (3, 3, 5, 5, 5, 5)? (b) There is a simple graph with 6 vertices with degree sequence (2, 3, 3, 4, 5, 5)?

Prove or disprove each of the following statements. (a) There exists a prime number x such...

Prove or disprove each of the following statements. (a) There exists a prime number x such that x + 16 and x + 32 are also prime numbers. (b) ∀a, b, c, m ∈ Z +, if a ≡ b (mod m), then c a ≡ c b (mod m). (c) For any positive odd integer n, 3|n or n 2 ≡ 1 (mod 12). (d) There exist 100 consecutive composite integers.

Prove or disprove the statements: (a) If x is a real number such that |x +...

Prove or disprove the statements: (a) If x is a real number such that |x + 2| + |x| ≤ 1, then x 2 + 2x − 1 ≤ 2. (b) If x is a real number such that |x + 2| + |x| ≤ 2, then x 2 + 2x − 1 ≤ 2. (c) If x is a real number such that |x + 2| + |x| ≤ 3, then x 2 + 2x − 1 ≤ 2....

Let U = {(x1,x2,x3,x4) ∈F4 | 2x1 = x3, x1 + x4 = 0}. (a) Prove...

Let U = {(x1,x2,x3,x4) ∈F4 | 2x1 = x3, x1 + x4 = 0}. (a) Prove that U is a subspace of F4. (b) Find a basis for U and prove that dimU = 2. (c) Complete the basis for U in (b) to a basis of F4. (d) Find an explicit isomorphism T : U →F2. (e) Let T as in part (d). Find a linear map S: F4 →F2 such that S(u) = T(u) for all u ∈...

How do you simplify the following boolean expressions: Y3= X3' X2 X1 X0' + X3' X2...

How do you simplify the following boolean expressions: Y3= X3' X2 X1 X0' + X3' X2 X1 X0 + X3 X2 X1 X0' +X3X2X1X0 Y2 = X3' X2 X1' X0' + X3' X2 X1' X0 + X3 X2 X1' X0' + X3 X2 X1' X0 + X3 X2' X1' X0' + X3 X2' X1' X0 + X3 X2' X1 X0' + X3 X2' X1 X0 Y1 = X3' X2' X1 X0' + X3' X2' X1 X0 + X3' X2 X1 X0'...

4.Maximize: Z = 2X1+ X2-3X3 Subject to: 2X1+ X2= 14 X1+ X2+ X3≥6 X1, X2, X3≥0...

4.Maximize: Z = 2X1+ X2-3X3 Subject to: 2X1+ X2= 14 X1+ X2+ X3≥6 X1, X2, X3≥0 Solve the problem by using the M-technique.

Prove E(X1 + X2 | Y=y) = E(X1 | Y=y) + E(X2 |Y=y). Prove both cases...

Prove E(X1 + X2 | Y=y) = E(X1 | Y=y) + E(X2 |Y=y). Prove both cases where all random variables are discrete and also when all random variables are continuous.

Prove or disprove if B is a proper subset of A and there is a bijection...

Prove or disprove if B is a proper subset of A and there is a bijection from A to B then A is infinite

Prove or disprove that the union of two subspaces is a subspace. If it is not...

Prove or disprove that the union of two subspaces is a subspace. If it is not true, what is the smallest subspace containing the union of the two subspaces.

(a) Consider three positive integers, x1, x2, x3, which satisfy the inequality below: x1 +x2 +x3...

(a) Consider three positive integers, x1, x2, x3, which satisfy the inequality below: x1 +x2 +x3 =17. (1) Let’s assume each element in the sample space (consisting of solution vectors (x1, x2, x3) satisfying the above conditions) is equally likely to occur. For example, we have equal chances to have (x1, x2, x3) = (1, 1, 15) or (x1, x2, x3) = (1, 2, 14). What is the probability the events x1 +x2 ≤8occurs,i.e.,P(x1 +x2 ≤8|x1 +x2 +x3 =17andx1,x2,x3 ∈Z+)(Z+...

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