Question 8 - A-Level Maths

Exam Board	AQA
Module	Further AS Paper 2 Discrete (Further AS Paper 2 Discrete)
Year	2020
Session	June
Marks	2
Topic	Groups

8 The set $S$ is defined as $$S = \{ a , b , c , d \}$$ Figure 2 shows a Cayley table for $S$ under the commutative binary operation \begin{table}[h]

\captionsetup{labelformat=empty} \caption{Figure 2}

\end{table} 8

1. Prove that there exists an identity element for $S$ under the binary operation
  [0pt] [2 marks]
  8
(ii) State the inverse of $b$ under the binary operation
8
Figure 3 shows a Cayley table for multiplication modulo 4 \begin{table}[h]
\captionsetup{labelformat=empty} \caption{Figure 3}
$\times _ { 4 }$ 0 1 2 3
0 0 0 0 0
1 0 1 2 3
2 0 2 0 2
3 0 3 2 1
\end{table} Mali says that, by substituting suitable distinct values for $a , b , c$ and $d$, the Cayley table in Figure 2 could represent multiplication modulo 4 Use your answers to part (a) to show that Mali's statement is incorrect.
\includegraphics[max width=\textwidth, alt={}, center]{21ed3b4e-a089-4607-b5d6-69d8aac03f31-20_2491_1736_219_139}