Questions: Binary Operations and Algebraic Structures

Question 1

Consider the set of odd integers under multiplication. Is this a binary operation? What about the set of even integers under multiplication?

• A. Odd integers: yes (closed); even integers: no (not closed, since 2×2=4 is not 'even enough')
• B. Odd integers: yes (odd×odd=odd, closed); even integers: yes (even×even=even, closed)
• C. Odd integers: no (odd×odd can be even); even integers: yes (closed)
• D. Neither set is closed under multiplication because multiplication is only defined on all integers

Answer: (B)

Closure requires that the operation on any two elements stays in the same set. Odd × odd is always odd (e.g., 3×5=15), so multiplication is a binary operation on odd integers. Even × even is always even (e.g., 2×4=8), so it's also closed. Checking closure is the first and non-trivial step — it is not automatic. Option C is wrong: an odd number times an odd number is always odd, never even.

Question 2

A student defines a structure (S, ★) and verifies that ★ is associative and that S contains an identity element, but finds that some elements have no inverse under ★. What is the strongest structure name that applies?

• A. Group — identity and associativity are sufficient
• B. Monoid — closure and identity are present; inverses are not required
• C. Magma — only closure can be assumed here
• D. Monoid — provided closure has also been verified

Answer: (D)

The hierarchy requires checking each axiom in order. A monoid requires: closure, associativity, and identity. The student has verified associativity and identity, but to name it a monoid, closure must also hold (a binary operation on S is defined only if the output is always in S). If closure is verified, then we have a monoid. If inverses also existed, it would be a group. The student can't assume closure — it must be explicitly checked. Option A is wrong because groups require inverses.

Question 3

Most set equipped with a binary operation automatically forms a group.

• T. True
• F. False

Answer: False

A group requires four properties: closure, associativity, existence of an identity element, and existence of inverses for every element. A set with a binary operation that is only closed is called a magma. Addition on the natural numbers ℕ, for instance, has closure and associativity and an identity (0), but most elements lack additive inverses (there is no natural number that adds to 3 to give 0), so ℕ under addition is a monoid, not a group.

Question 4

The integers ℤ under addition form a group.

• T. True
• F. False

Answer: True

All four group axioms hold: (1) Closure — the sum of any two integers is an integer. (2) Associativity — (a+b)+c = a+(b+c) for all integers. (3) Identity — 0 is an identity element, since 0+a = a+0 = a. (4) Inverses — every integer n has an additive inverse −n, since n+(−n) = 0. This is the canonical example of a group and should be the reference case for checking the axioms.

Question 5

Why is checking closure the essential first step before classifying any algebraic structure, and why is it easy to overlook?

Model answer: Closure requires that the operation on any two elements of the set produces a result that is also in that set. It is easy to overlook because we often work with sets (like all integers or all real numbers) where the natural operation is obviously closed. But many interesting sets have operations that are not closed: subtraction on natural numbers produces negative numbers (outside ℕ), the inverse of an integer under multiplication is a fraction (outside ℤ), and odd integers under addition give even numbers. Without closure, the structure doesn't even qualify as a binary operation on that set, making all further classification meaningless.

The deeper reason closure matters is that all subsequent axioms (associativity, identity, inverses) are only meaningful if the operation is already closed — otherwise you might apply the operation and immediately leave the set, making terms like 'identity element' undefined. Experienced mathematicians sometimes skip explicitly stating closure because it seems obvious for familiar examples, which is exactly why beginners mistakenly skip verifying it on unfamiliar ones.