Abstract Algebra Answers

Prove or disprove that 2 + (11)^1/2∈Q .

Check whether the sequence, {Sn} where

Sn=1/1!+1/3!+1/5!+....1/(2n+1)!
Is convergent or not.

Show that the roots of the equation
Z^4-1=0
form a cyclic group of order 4

Given that x*y=1+x+y, show wheather the binary operation is associative algebrically

Prove or disprove:
a) every left simple ring is right simple
b) every simple ring is left simple

Find all primes p for which −5 is a quadratic residue.

Consider the following sets together with binary operations.
Are they user-friendly? Z with binary operation
z1 . z2 = 2z1 - 4z2
Is the set closed under the operation? Is the operation commutative ?
Is the operation associative ? Is there an identity? If there is an identity
element then does every element have an inverse relative to the operation _
Consider R together with x . y = x/y. Ask the same questions as in last
example.

Prove or disprove that C~ R as fields.

1. which of the following statement is true
a. \\(\\sim (p\\vee q)=\\sim (p\\wedge \\sim q)\\)
b. \\((p\\vee q)\\wedge (p\\vee r)=p\\vee (q\\wedge r)\\)
c. \\((p\\wedge q)\\wedge (p\\vee r)=p\\vee (q\\wedge r)\\)
d. \\((p\\wedge q)\\vee (p\\vee r)=p\\vee (q\\wedge r)\\)

2. ____ reads “the goods are standard if and only if the goods are expensive”
a. \\(\\sim (\\sim p\\wedge \\sim q)\\)
b. \\(\\sim \\sim q\\)
c. \\(p\\leftrightarrow q\\)
d. \\(\\sim p\\wedge q\\)