If the operation * is defined for all integers a and b by a*b=a+b-ab, which of the following statements must be true for all intergers a, b and c?
a 1 only
b II only
c I and II only
d I and III only
e I, II and III
I: distributive property: a+b-ab = b+a -ba same thing so I is good.
II a+0-a(0) = a II is good
If both sides cancel then they equal each other:
III: a+b-ab*c = a*b+c-bc --> a+b-ab+c -(a+b-ab)c=b+c-bc+a -(b+c-bc)a -->
a+b-ab+c-ac-bc+abc=b+c-bc+a -ab -ac +abc indeed both sides cancel so III is also sufficient.