Further Mathematics: Sets, relations and Groups Topical Past Paper Questions

(a) The relation $R$ is defined such that $aRb$ if and only if $p$ divides $a^2 - b^2$, where $p$ is a fixed prime number greater than 2 and $a, b \in \mathbb{Z}^+$. (i) Show that R is an equivalence relation. (ii) Find, in terms of p, the two smallest positive integers that are…

(a) Consider the set $G=\{1,2,3,4,5,6\}$ and the operation $x_{7}$ which denotes multiplication modulo 7. (i) Copy and complete the following Cayley table for $G$. (ii) Show that $\{G, x_{7}\}$ is a group. (iii) Show that $\{G, x_{7}\}$ is cyclic. 0 (b) The groups A and B are bot…

[Maximum mark: 19] The set of all integers from 0 to 99 inclusive is denoted by S. The binary operations $ * $ and $ \circ $ are defined on S by $$ a*b=[a+b+20](\bmod100) $$ $$ a\circ b=[a+b-20](\bmod100). $$ (a) Find the identity element of S with respect to $ * $. [3] (b) Show…