TSSFL TECHNOLOGY STACK

To prove that a binary operation on a certain set, say makes it a group, we need to show that four properties are satisfied. Let's illustrate this by examples.

Problem 1

Let be a set of non-zero rational numbers such that , . Show that is a group.

(i) Closure: If and are non-zeros, then is non-zero rational number, and so

(ii) Associative: For



Similarly,



(iii) Identity element: The number two acts as the multiplicative identity, so,



(iv) Inverse: If is non-zero, then is a no-zero rational number and serves as the multiplicative inverse of , since



Problem 2

Let , i.e., the set of all real numbers greater than . Define

for Show that the operation makes a group.

Solution

For to be a group, four properties: closure, associative, identity element and inverse must be satisfied.

(i) Closure: If , then so and therefore multiplying by both sides, it follows immediately that



Hence the operation is closed on .

(ii) Associative: For we have



On the other hand, we have



Thus



(iii) Identity: Hence, 2 is the identity element for the operation .

(iv) Inverse: Given any we need to solve the equation This gives the equation



This solution belongs to since

NB: You can also check that