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\\)

1. ____is equivalent to \\((p\\vee q)\\)
a. \\(\\sim (\\sim p\\wedge \\sim q)\\)
b. \\((\\sim \\sim q)\\)
c. \\((p\\leftrightarrow q)\\)
d. \\((\\sim p\\wedge q)\\)
2. \\((p\\vee q)\\vee r=p\\vee (q\\vee r)\\) and \\((p\\wedge q)\\wedge r = p\\wedge (q\\wedge r)\\) implies an ___
a. Distributive Laws
b. Commutative Laws
c. Associative laws
d. Idempotent Laws

Let d ∈N , where d ≠1 and d is not divisible by the square of a prime. Prove that N:Z[square root of d] maps N union {0} : N(a+b sqr root d) = |a^2 -db^2| satisfies the following properties for x,y belongs to Z[sqr root d]. 1. N(x) = 0 if x=0. 2 N(xy) = N(x)N(y) 3. N(x) =1 if x is a unit 4. N(x) is prime if x is irreducible.

If p*q = p^2-q^2-2pq. Find the inverse of p under the operation.

State and prove generalized commutative law in a commutative semigroup

Q. Find the dimension of the subspace of R4 that is span of the vectors
(█(1¦(-1)@0@1)), (█(2¦1@1@1)),(█(0¦0@0@0)),(█(1¦1@-2@-5))
Q. Choose the correct answer.
Q. Let b and c are elements in a group G and e is identity element of G. If b5=c3=e,then inverse of bcb2 is
a. b2cb
b. b3c2b4
d. b2c2b4

Let N ∈d , where d ≠1 and d is not divisible by the square of a prime. Prove that N:Z[square root of d] maps N union {0} : N(a+b sqr root d) = |a^2 -db^2| satisfies the following properties for x,y belongs to Z[sqr root d].
1. N(x) = 0 if x=0.
2 N(xy) = N(x)N(y)
3. N(x) =1 if x is a unit
4. N(x) is prime if x is irreducible.