Question

"The sum of the first three terms of a finite geometrical series is -7/10 and their product is -1/125. [Hint: use a/r, a and ar to represent the first three terms, respectively.] The three numbers are ____,_____ and ____.

Answer 1

Answer:

NOTES ON GROUP THEORY

1. Binary Structure

Let S be a set. We denote by S × S the set of ordered pairs (a, b), where

a, b ∈ S. Thus the ordered pairs (a, b) and (b, a) represent distinct elements

of S × S unless a = b.

A binary operation ? on S is a function from S × S into S. Thus for every

(a, b) ∈ S × S, the binary operation ? assigns a unique element a ? b of S. If

this happens, then we say that the pair (S, ?) is a binary structure.

Let us understand the above notion through examples.

Example 1.1 : We follow the standard notations to denote the set of natural numbers, integers, rationals, reals, complex numbers by N, Z, Q, R, C

respectively. If S is one of the sets above, then S

∗

stands for S \ {0}.

(1) Addition (resp. multiplication) is a binary operation on Z (resp. Q).

(2) Division is not a binary operation on Z

∗

.

(3) Subtraction is a binary operation on Z but not on N.

(4) Division is a binary operation on R

∗

(resp. C

∗

).

As seen in (3), it may happen that a ? b /∈ A for some a, b ∈ A.

Let (S, ?) be a binary structure. Let A be a subset of a set S. We say

that ? is an induced binary operation on A if a ? b ∈ A for every a, b ∈ A.