#7
7 Prove or disprove: If H is a normal subgroup of G such that H and G/H are abelian, then G is abelian. If G is cyclic, prove that G/H must also be cyclic. 8.
2. Consider Z7 Prove that the operation
on Z7 dened by [x]7
[y]7 =
[5xy]7 is well dened.
= 2. Consider Z- Prove that the operation ♡ on Z- defined by [2]7 0 [y]; [5xy]7 is well defined.
Prove using mathematical induction:
(4) Prove that for all n E N, 3(7" – 4”).
Prove or disprove that 7n=O(5n) + 7.
# 7 please
6. Prove that if x is rational and y is irrational, then 2 +y is irrational. 7. Prove that if x, y € R+ such that Ty # #4, then x + y.
5. Prove the folowing divergence property div(4a,-(grade)-a + φ div a 6. Prove 7. Compute 8. Demonstrate that
7.) State the Fundamental Theorem of Arithmetic and use it to
prove that
3 p
625 is irrational.
7.) State the Fundamental Theorem of Arithmetic and use it to prove that 625 is irrational.
Three questions!please!
7. Prove that J(x) is integrable on (0,b), and calculate their integral. 8. Prove that the following function is integrable on [0, 1], and calculate the integral. 1 if for some n E N 0 (z)= otherwise. 8. Prove that if f is integrable on (a, b, then f2 is also integrable on la,b
5. (a) (7 points) Use the definition of convergence to prove that the sequence {(-1)-+ 히 converges to 0 (b) (7 points) Prove that the sequence k=1 does not converge.
7. Matrix A is said to be involutory if A² = 1. Prove that a square matrix A is both orthogonal and involutory if and only if A is symmetric.