1. Z with binary operation z1⋅z2=2z1−4z2 .
The set Z is closed under ⋅ because if z1,z2∈Z , then 2z1∈Z , 4z2∈Z , 2z1−4z2∈Z .
The operation ⋅ is not commutative because
0⋅1=2×0−4×1=−4=2=2×1−4×0=1⋅0. The operation ⋅ is not associative because
(1⋅0)⋅0=2⋅0=4=2=1⋅0=1⋅(0⋅0). If z is an identity, then 0=0⋅z=−4z and 1=1⋅z=2−4z . Subtracting the first equation from the last gives 1=2 , contradiction. Therefore, there is no identity.
2. R with binary operation x⋅y=x/y .
The set R is closed under ⋅ because if x,y∈R , then x/y∈R . Also ⋅ is a partial operation because x⋅0 is undefined for all x∈R .
The operation ⋅ is not commutative because
1⋅2=1/2=2=2/1=2⋅1. The operation ⋅ is not associative because
(1⋅1)⋅2=(1/1)/2=1/2=2=1/(1/2)=1⋅(1⋅2). If x is an identity, then 1=x⋅1=x and 2=x⋅2=x/2 . Dividing the last equation by the first gives 2/1=(x/2)/x=1/2 , contradiction. Therefore, there is no identity.
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