Answer to Question #97340 in Abstract Algebra for Brendan

Question #97340
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.
1
Expert's answer
2019-10-27T13:38:57-0400

1. Z\mathbb{Z}  with binary operation z1z2=2z14z2z_1\cdot z_2 = 2z_1 - 4z_2 .

The set Z\mathbb{Z}  is closed under \cdot  because if z1,z2Zz_1, z_2\in \mathbb{Z} , then 2z1Z2z_1\in \mathbb{Z} , 4z2Z4z_2\in \mathbb{Z} , 2z14z2Z2z_1 - 4z_2\in \mathbb{Z} .

The operation \cdot  is not commutative because

01=2×04×1=42=2×14×0=10.0\cdot 1 = 2\times 0 - 4\times 1 = -4 \neq 2 = 2\times 1 - 4\times 0 = 1\cdot 0.

The operation \cdot  is not associative because


(10)0=20=42=10=1(00).(1\cdot 0)\cdot 0 = 2\cdot 0 = 4 \neq 2 = 1\cdot 0 = 1\cdot (0\cdot 0).

If zz  is an identity, then 0=0z=4z0 = 0\cdot z = -4z  and 1=1z=24z1 = 1\cdot z = 2 -4z . Subtracting the first equation from the last gives 1=21=2 , contradiction. Therefore, there is no identity.


2. R\mathbb{R}  with binary operation xy=x/yx\cdot y = x/y .

The set R\mathbb{R}  is closed under \cdot  because if x,yRx, y\in \mathbb{R} , then x/yRx/y\in \mathbb{R} . Also \cdot  is a partial operation because x0x\cdot 0  is undefined for all xRx\in \mathbb{R} .

The operation \cdot  is not commutative because


12=1/22=2/1=21.1\cdot 2 = 1/2 \neq 2 = 2/1 = 2\cdot 1.

The operation \cdot  is not associative because


(11)2=(1/1)/2=1/22=1/(1/2)=1(12).(1\cdot 1)\cdot 2 = (1/1)/2 = 1/2 \neq 2 = 1/(1/2) = 1\cdot (1\cdot 2).

If xx  is an identity, then 1=x1=x1 = x\cdot 1 = x  and 2=x2=x/22 = x\cdot 2 = x/2 . Dividing the last equation by the first gives 2/1=(x/2)/x=1/22/1 = (x/2)/x = 1/2 , contradiction. Therefore, there is no identity.


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