1. A binary operation β is defined on a set of positive real numbers by π¦ β π₯ = π¦ % + π₯ % + ππ¦ ( + ππ₯ ( β 5π¦ β 5π₯ + 16 where π β π . (i) State giving a reason for your answer, if βis commutative. (ii) Given that π π₯ = 2 β π₯ and π₯ β 1 is a factor of π(π₯). (a) Find a (b) factorize π(π₯) completely. 2. A binary operation β is defined on a set of positive real numbers by π¦ β π₯ = π¦ ( + π₯ ( + 2π¦ + π₯ β 5π₯π¦ = 0 Solve the equation 2 β π₯ = 0. 3. The set S consist of all numbers of the form π + π 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 π 56 , the set of all non-negative real numbers by π₯ β π¦ = ln(π ; + π < β 1) (i) (ii) (iii) Show that π 56 is closed with respect to *. Show that there is an identity in π 56 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 π₯ β π¦ = (π₯ % + π¦ % )> . 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 π₯, π¦, π β π . An operation * is defined by π₯ β π¦ = * is associative. ;< ;6<AB where π₯ + π¦ β π. Show that