Question

1. Find out whether or not the integer pairs are relatively prime: (8, 15), (6, 50), (3, 31) and (3, 21).

2. Is the set of all real numbers under the arithmetic addition and multiplication a field? Justify your answer.

3. Consider a set S={a,b} with addition and multiplication defined by: a+a=a, a+b=b, b+a=b, b+b=a, axa=a, axb=a, bxa=a, bxb=b. Is S a ring? Justify your answer.

Answer 1

If GCF is 1 then numbers are co-prime:

1)

1. (8,15) GCF is 1 so these are coprime

2. (6,50) Not coprime

3. (3,31) CO-PRIME

    

4. (3,21) NOT COPRIME

2)

For field following properties should hold:

So we know that real numbers are associative under addition.

So we know that addition operation is communtative for real numbers

For real numbers additive identity is 0.

For every real number a there exist –a.

For real numbers multiplication is associative.

For real numbers multiplication is commutative as well.

For real numbers multiplicative identiy is 1.

For every real number a there exist 1/a such that a.(1/a) =1 except additive identity i.e 0

Real numbers are also distributive.

So set of all real numbers under the arithmetic addition and multiplication a field.

Ques 3:

Now all operations a+a=a, a+b=b, b+a=b, b+b=a, axa=a, axb=a, bxa=a, bxb=b are closed under multiplication and addition.

LHS

a + (b + c)

if c = a

a + (b + a) = a + b = b                              ….(1)

if c = b

a + (b + b) = a + a =a                              ….(2)

RHS

(a + b) + c

if c = a

(a + b )+ a = b + a = b                              ….(3)

if c = b

(a + b) + b = b + b =a                                ….(4)

AS 1 = 2 and 3 = 4 Hence it is associative

Here zero element is a as

a + a = a

and b + a = b

Here inverse of a is a such that a+a = a (zero element)

And inverse of b is b as b + b = a (zero element)

a + b = b

b + a = b

hence it is commutative

LHS

a.(b.c)

if c =a

a.(b.a) = a.a=a          …(1)

if c= b

a.(b.b) = a.b=a          …(2)

RHS

if c =a

(a.b).a = a.a=a          …(3)

if c= b

(a.b).b = a.b=a          …(4)

1 = 3 and 2 = 4 hence multiplication is associative.

a.(b+c) = a.b + a.c

let c=a

a.(b+a) = a.b +a.a

a.b = a + a

a = a

Its true for c = a

Let c = b

a.(b+b) = a.b + a.b

a.a=a+a

a = a

hence its true for c = b as well.

(a+b).c = a.b +b.c

Let c =a

(a+b).a=a.b+b.a

b.a = a+a

a = a

hence its true for c = a

(a+b).c = a.b +b.c

Let c =b

(a+b).b=a.b+b.b

b.b=a+b

b=b

hence its true for c =b

hence this property also holds and S is a ring