Answer to Question #275052 in Discrete Mathematics for adie

Question

Let the operations * and ⊕ be defined on the set of integers.

Find each of the following are commutives, associatives, left-distributive, and right-distributive.

1.   a*b=ab; a⊕b=-a-b

2.   a*b=a+2b; a⊕b=a-2b

3.   a*b=2ab; a⊕b=2a+2b+4ab

4.   a*b=-a-b-2ab; a⊕b=3a+3b

5.   x*y=x²+2x+y²; x⊕y=x+y

Answer 1

1)* is commutative and associative, ⊕ is not commutative because a-b is not equal to b-a and not associative, since (a⊕b)⊕c=(a-b)-c=a-b-c while a⊕(b⊕c)=a-(b-c)=a-b+c. * is left- and right-distributive over ⊕ since * is commutative and a*(b⊕c)=a(b-c)=ab-ac=(a*b)⊕(a*c). ⊕ is not left- or right-distributive over *, because (a⊕b)*(a⊕c)=(a-b)(a-c)=aa-ba-ac+bc and (b⊕a)*(c⊕a)=(b-a)(c-a)=aa-ba-ac+bc while a⊕(b*c)=a-bc and (b*c)⊕a=bc-a.

2)* is not commutative and not associative, ⊕ is also not commutative and not associative because in both cases aR(bRc) resulting in 4c while (aRb)Rc give only 2c. a*(b⊕c)=a+2(b-2c)=a+2b-4c, (a*b)⊕(a*c)=(a+2b)-2(a+2c)=a+2b-2a-4c=-a+2b-4c, (b⊕c)*a=b-2b +2a, (b*a)⊕(c*a)=(b+2a)-2(c+2a)=b+2a-2c-4a=b-2a-2c, so * is not left- or right-distributive over ⊕. a⊕(b*c)=a-2(b+2c)=a-2b+4c, (a⊕b)*(a⊕c)=a-2b+2a-4c=3a-2b-4c, (b*c)⊕a=b+2c-4a, (b⊕a)*(c⊕a)=b-2a+2c-4a=b+2c-6a, so ⊕ is not left- or right-distributive over *.

3)* is commutative and associative, ⊕ is commutative and not associative because a⊕(b⊕c)=2a+4b+4c+8bc+8ac+8ab+16abc and (a⊕b)⊕c=4a+4b+8ab+2c+8ac+8bc+16abc. a*(b⊕c)=2a(2b+2c+4bc)=4ab+4ac+8abc, (a*b)⊕(a*c)=2(2ab)+2(2ac)+4(2ab2ac)=4ab+4ac+16a²bc, the same with another side, so * is not left- or right-distributive over ⊕. a⊕(b*c)=2a+4bc+8abc, (a*b)⊕(a*c)=4ab+4ac+16a²bc, also the same for other side so ⊕ is not left- or right-distributive over *.

4)* is commutative and not associative cause after using * twice argument will retain sign, and change if using once, ⊕ is commutative and not commutative because pair in parentheses will be multiplied by 9 while outside member will be multiplied by 3. a*(b⊕c)=a*(3b+3c)=-a-3b-3c-6ab-6ac, (a*b)⊕(a*c)=(-a-b-2ab)⊕(-a-c-2ac)=-6a-3b-3c-6ab-6ac, since * is commutative, we don't need to show, what * is not left- or right-distributive over ⊕. a⊕(b*c)=3a-3b-3c-6bc, (a⊕b)*(a⊕c)=-3a-3b-3a-3c-2(3a +3b)(3a+3c)=-6a-3b-3c-18a²-18ab-18ac-18bc, ⊕ also is commutative, so ⊕ is not left- or right-distributive.

5) * is not commutative and not associative, ⊕ is commutative and associative. a*(b⊕c)=a²+2a+b²+2bc+c², (a⊕b)*(a⊕c)=a²+2ab+b²+2a+2b+a²+2ac+c², so * is not left-distributive over ⊕, (b⊕c)*a=b²+2bc+c²+2b+2c+a², (b⊕a)*(c⊕a)=b²+2ba+a²+2b+2a+c²+2ca+a², so * is not right-distributive over ⊕. a⊕(b*c)=a+b²+2b+c², (a*b)⊕(a*c)=a²+2a+b²+a²+2a+c², ⊕ is commutative, so ⊕ is not left- or right-distributive over *.