1. A binary operation ⊕ is defined on a set of positive real numbers by
𝑦 ⊕ 𝑥 = 𝑦 % + 𝑥 % + 𝑎𝑦 ( + 𝑎𝑥 ( − 5𝑦 − 5𝑥 + 16 where 𝑎 ∈ 𝑅.
(i)
State giving a reason for your answer, if ⊕is commutative.
(ii)
Given that 𝑓 𝑥 = 2 ⊕ 𝑥 and 𝑥 − 1 is a factor of 𝑓(𝑥).
(a) Find a (b) factorize 𝑓(𝑥) completely.
2. A binary operation ⊕ is defined on a set of positive real numbers by
𝑦 ⊕ 𝑥 = 𝑦 ( + 𝑥 ( + 2𝑦 + 𝑥 − 5𝑥𝑦 = 0
Solve the equation 2 ⊕ 𝑥 = 0.
3. The set S consist of all numbers of the form 𝑎 + 𝑏 5 where a and b are integers. Show
that
(i)
(ii)
(iii)
S is closed under addition and multiplication.
There is an identity in S for addition and also an identity in S for multiplication.
Not every element of S has an inverse with respect to multiplication.
4. The binary operation * is defined on 𝑅56 , the set of all non-negative real numbers by
𝑥 ∗ 𝑦 = ln(𝑒 ; + 𝑒 < − 1)
(i)
(ii)
(iii)
Show that 𝑅56 is closed with respect to *.
Show that there is an identity in 𝑅56 with respect to *
Show that there is only one element, which is to be found, that has an inverse with
respect to *.
=
5. The binary operation * is defined on R by 𝑥 ∗ 𝑦 = (𝑥 % + 𝑦 % )> . Show that (i) * is
associative (ii) there is an identity element with respect to * (iii) every element of R has
an inverse with respect to * (iv) the operation * is not distributive over +.
6. Let 𝑥, 𝑦, 𝑘 ∈ 𝑅. An operation * is defined by 𝑥 ∗ 𝑦 =
* is associative.
;<
;6<AB
where 𝑥 + 𝑦 ≠ 𝑘. Show that