Question
Download Solution PDFConsider the set G = {a + b
Answer (Detailed Solution Below)
Detailed Solution
Download Solution PDFSolution:
The set G = {a + b\(\sqrt2\) : a, b ∈ Q, the set of all rational numbers} with respect to binary operation usual addition.
Satisfies closure, associative, identity, inverse, and commutative properties.
(1) Closure axiom:
- Let x,y ∈ G. Then
-
- Since (a + c) and (c + d) are rational numbers.
- ∴ G is closed with respect to addition.
(2) Associative axiom:
- Since the elements of G are real numbers, addition is associative.
(3) Identity axiom:
- There exists
such that for all -
- Similarly, we have 0 + x = x.
- ∴ 0 is the identity element of G and satisfies the identity axiom.
(4) Inverse axiom:
- For each
, there exists such that - =
- Similarly, we have (-x) + x = 0.
- ∴
is the inverse of and satisfies the inverse axiom.
(5) Commutative axiom:
-
- =
- = y + x, for all x,y ∈ G.
- ∴ The commutative property is true.
Hence, Non-commutativity property fails for G
Last updated on May 26, 2025
-> AAI ATC exam date 2025 will be notified soon.
-> AAI JE ATC recruitment 2025 application form has been released at the official website. The last date to apply for AAI ATC recruitment 2025 is May 24, 2025.
-> AAI JE ATC 2025 notification is released on 4th April 2025, along with the details of application dates, eligibility, and selection process.
-> Total number of 309 vacancies are announced for the AAI JE ATC 2025 recruitment.
-> This exam is going to be conducted for the post of Junior Executive (Air Traffic Control) in Airports Authority of India (AAI).
-> The Selection of the candidates is based on the Computer Based Test, Voice Test and Test for consumption of Psychoactive Substances.
-> The AAI JE ATC Salary 2025 will be in the pay scale of Rs 40,000-3%-1,40,000 (E-1).
-> Candidates can check the AAI JE ATC Previous Year Papers to check the difficulty level of the exam.
-> Applicants can also attend the AAI JE ATC Test Series which helps in the preparation.