Prove that \(f\) is

(i)     not injective;

(ii)     not surjective.

The relation \(R\) is defined for \(a,{\text{ }}b \in \mathbb{R}\) so that \(aRb\) if and only if \(f(a) \times f(b) = 1\).

Show that \(R\) is an equivalence relation.

The relation \(R\) is defined for \(a,{\text{ }}b \in \mathbb{R}\) so that \(aRb\) if and only if \(f(a) \times f(b) = 1\).

State the equivalence classes of \(R\).

(i)     eg\(\;\;\;f(2) = f(3)\)     M1

hence \(f(a) = f(b) \Rightarrow a = b\)     R1

so not injective     AG

(ii)     eg\(\;\;\;\)Codomain is \(\mathbb{R}\) and range is \(\{  - 1,{\text{ }}1\} \)     M1

these not the same so not surjective     R1AG

Note:     if counter example is given it must be stated it is not in the range to obtain the R1. Eg \(f(x) = 2\) has no solution as \(f(x) \in \{  - 1,{\text{ }}1\} \forall x\).

[4 marks]

if \(a \ge 0\) then \(f(a) \times f(a) = 1 \times 1 = 1\)     A1

if \(a < 0\) then \(f(a) \times f(a) =  - 1 \times  - 1 = 1\)     A1

in either case \(aRa\) so \(R\) is reflexive     R1

\(aRb \Rightarrow f(a) \times f(b) = 1 \Rightarrow f(b) \times f(a) = 1 \Rightarrow bRa\)     A1

so \(R\) is symmetric     R1

if \(aRb\) then either \(a \ge 0\) and \(b \ge 0\) or \(a < 0\) and \(b < 0\)

if \(a \ge 0\) and \(b \ge 0\) and \(bRc\) then \(c \ge 0\) so \(f(a) \times f(c) = 1 \times 1 = 1\) and \(aRc\)     A1

if \(a < 0\) and \(b < 0\) and \(bRc\) then \(c < 0\) so \(f(a) \times f(c) =  - 1 \times  - 1 = 1\) and \(aRc\)     A1

in either case \(aRb\) and \(bRc \Rightarrow aRc\) so \(R\) is transitive     R1

Note:     Accept

\(f(a) \times f(b) \times f(b) \times f(c) = 1 \times 1 = 1 \Rightarrow f(a) \times 1 \times {\text{ }}f(c) = 1 \Rightarrow {\text{ }}f(a) \times f(c) = 1\)

Note:     for each property just award R1 if at least one of the A marks is awarded.

as \(R\) is reflexive, symmetric and transitive it is an equivalence relation     AG

[8 marks]

equivalence classes are \([0,{\text{ }}\infty [\) and \(] - \infty ,{\text{ }}0[\)     A1A1

Note:     Award A1A0 for both intervals open.

[2 marks]

Total [14 marks]