When is a binary operation closed on a set?

When the result of combining any two elements of the set is always in the set.

Test for commutativity?

a\ast b=b\ast a for all a and b.

Test for associativity?

(a\ast b)\ast c=a\ast(b\ast c) for all a,b,c.

How do you prove an operation is commutative?

Show with general a,b that a\ast b and b\ast a simplify to the same expression.

How do you prove an operation is associative?

Expand (a\ast b)\ast c and a\ast(b\ast c) separately and show they are equal.

Why use unequal values for a commutativity counter-example?

If a=b then swapping them changes nothing.

How is x\otimes y (multiplication modulo n) calculated?

Multiply, then take the remainder on division by n.

Is matrix multiplication commutative?

Not in general: AB\neq BA for most matrices. It is associative.

In a Cayley table, which element is first in a\circ b?

The element on the left (the row).

How do you see closure in a Cayley table?

Every entry is an element of the set.

How do you see commutativity in a Cayley table?

The table is symmetric about the leading diagonal.

Binary operations and their propertiesAQA A-Level Further Maths: Flashcards

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What is a binary operation on a set?

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What is a binary operation on a set?: A rule that combines any two elements of the set to give one element.
When is a binary operation closed on a set?: When the result of combining any two elements of the set is always in the set.
Test for commutativity?: $a\ast b=b\ast a$ for all $a$ and $b$ .
Test for associativity?: $(a\ast b)\ast c=a\ast(b\ast c)$ for all $a,b,c$ .
How do you prove an operation is commutative?: Show with general $a,b$ that $a\ast b$ and $b\ast a$ simplify to the same expression.
How do you prove an operation is associative?: Expand $(a\ast b)\ast c$ and $a\ast(b\ast c)$ separately and show they are equal.
How do you disprove commutativity or associativity?: Give one counter-example.
Why use unequal values for a commutativity counter-example?: If $a=b$ then swapping them changes nothing.
How is $x\otimes y$ (multiplication modulo $n$ ) calculated?: Multiply, then take the remainder on division by $n$ .
Is matrix multiplication commutative?: Not in general: $AB\neq BA$ for most matrices. It is associative.
In a Cayley table, which element is first in $a\circ b$ ?: The element on the left (the row).
How do you see closure in a Cayley table?: Every entry is an element of the set.
How do you see commutativity in a Cayley table?: The table is symmetric about the leading diagonal.

Exam questions on Binary operations and their properties

The binary operation $\ast$ is defined on the set of real numbers by $a\ast b=2a+3b$ .
Show that $\ast$ is not associative.2 marks
The binary operation $\otimes$ is defined on the set $\{0,1,2,3,4,5\}$ by $x\otimes y=$ the remainder when $xy$ is divided by 6, that is multiplication modulo 6.
Prove that $\otimes$ is commutative.2 marks
The binary operation $\circ$ is defined on the set $S=\{1,2,3,4\}$ by $a\circ b=$ the remainder when $ab$ is divided by 5.
Construct the Cayley table for $\circ$ on $S$ , with the first element of $a\circ b$ given by the row.3 marks

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Written by the Exaim team, led by Shaun Daswani (Head of Upper Secondary, Improve ME Institute; MSc Financial Mathematics, Imperial College London; BSc, UCL) and Jason Daswani (operational lead, Improve ME Institute; LSE).