********************SOLUTION TO THE MATH QUIZ ********************** We want to prove the following statement with the method of induction : In a finite group of people where one of them is blond, all the people are blonds - Basis : If the group has only one member who is blond, all the group is blond. - Inductive step n --> n+1 : Suppose that the hypothesis has been proved for groups with 2,3,4,..., n members. We have to show that the hypothesis for n+1 members is also true. If we have a group with n+1 members, we can split it into 2 groups with 2 (included the blond person) and n -1 members. Because of the inductive hypothesis, the group with 2 members is blond. We can thus add one of the blond members (of the group with 2 members) to the group with n - 1 members, which will lead to a group with n members. Therefore, because of the inductive hypothesis, the group with n members is also blond. Therefore, the entire group with n+1 members is blond. Since both the basis and the inductive step have been performed, by mathematical induction, the statement is true. Find the error. ------------------------------------------------------------ In the statement we define a group of people, where in mathematics the definition of a group, written G, is a set of elements together with an operation · (the group law of G, and where the symbol • is a symbol which represents a given operation, like for instance addition or multiplication), which combines any 2 elements a and b to form a third element ab (or a • b). A group has to satisfy the group axioms (otherwise its not a group), namely closure, associativity, identity element and inverse element. Closure : For all a, b in G, a • b must be also in G Associativity: For all a,b in G, (a • b) • c = a • (b • c) Identity element: There exists an element e in G such that for every element a in G, e • a = a • e = a Inverse element: For each a in G, there exists an element b in G such that a • b = b • a = e, where e is the identity element (for groups called abelian groups the commutativity equation a • b = b • a always holds; for instance in the the group of integers under addition, the commutativity equation a + b = b + a always holds; however it is not always true as it depends on the operation (and operands). Now, a group must contain at least one element. In that case, it is called trivial group or identity group, and it is the unique (up to isomorphism) group containing e which is the identity element. In our statement we state that we have a finite group of blond people, since we say that if one of them is blond, all the people are blond; in the basis step, we want to prove the statement for n = 1, that is only one element (member) which is blond. However, since being blond is a specific property, the blond person is not an identity element. Therefore, the statement is not true for 1 element (at the basis step) because a group with only one blond person, which is not the identity element, cannot exist (its simply not a group).
Posted on: Sun, 21 Sep 2014 22:49:21 +0000