Definition: inverse of a?

a^{-1}\in S with aa^{-1}=a^{-1}a=e.

Why does 0 have no inverse under a\circ b=\frac{ab}{4}?

0\circ x=0 for all x, which is never the identity 4.

How do you find an inverse of a?

Solve ax=e for x, then check x\in S.

What must be true of an identity found by solving ae=a?

It does not depend on a, it is in S, and ea=a also holds.

Where do you find the identity in a Cayley table?

The row and column that repeat the headings unchanged.

Identity and inversesAQA A-Level Further Maths: Flashcards

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Definition: identity element $e$ of $(S,*)$?

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Definition: identity element $e$ of $(S,*)$ ?: $a*e=e*a=a$ for every $a\in S$ .
Definition: inverse of $a$ ?: $a^{-1}\in S$ with $a*a^{-1}=a^{-1}*a=e$ .
Identity of $a*b=a+b-3$ ?: $3$
Inverse of $a$ under $a*b=a+b-3$ ?: $6-a$
Identity of $a\circ b=\frac{ab}{4}$ ?: $4$
Identity of $a*b=a+b+ab$ on $\mathbb{R}\setminus\{-1\}$ ?: $0$
Inverse of $a$ under $a*b=a+b+ab$ ?: $-\frac{a}{1+a}$
Why does $0$ have no inverse under $a\circ b=\frac{ab}{4}$ ?: $0\circ x=0$ for all $x$ , which is never the identity $4$ .
How do you find an inverse of $a$ ?: Solve $a*x=e$ for $x$ , then check $x\in S$ .
What must be true of an identity found by solving $a*e=a$ ?: It does not depend on $a$ , it is in $S$ , and $e*a=a$ also holds.
Inverse of $3$ under multiplication modulo $7$ ?: $5$ , since $3\times5=15\equiv1$ .
Where do you find the identity in a Cayley table?: The row and column that repeat the headings unchanged.

Exam questions on Identity and inverses

The binary operation $*$ is defined on the set of real numbers $\mathbb{R}$ by $a*b=a+b-3$ .
Find the element of $\mathbb{R}$ that is its own inverse.2 marks
The binary operation $\circ$ is defined on the set of real numbers $\mathbb{R}$ by $a\circ b=\frac{ab}{4}$ .
Explain why the element $0$ has no inverse in $\mathbb{R}$ under $\circ$ .2 marks
The set $S=\{1,2,3,4,5,6\}$ under the operation $\otimes$ , multiplication modulo 7.
Show that $1$ is the identity element and find the inverse of each element of $S$ .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).