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Binary Operations Exercise Sheet

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1. A binary operation βŠ• is defined on a set of positive real numbers by
𝑦 βŠ• π‘₯ = 𝑦 % + π‘₯ % + π‘Žπ‘¦ ( + π‘Žπ‘₯ ( βˆ’ 5𝑦 βˆ’ 5π‘₯ + 16 where π‘Ž ∈ 𝑅.
(i)
State giving a reason for your answer, if βŠ•is commutative.
(ii)
Given that 𝑓 π‘₯ = 2 βŠ• π‘₯ and π‘₯ βˆ’ 1 is a factor of 𝑓(π‘₯).
(a) Find a (b) factorize 𝑓(π‘₯) completely.
2. A binary operation βŠ• is defined on a set of positive real numbers by
𝑦 βŠ• π‘₯ = 𝑦 ( + π‘₯ ( + 2𝑦 + π‘₯ βˆ’ 5π‘₯𝑦 = 0
Solve the equation 2 βŠ• π‘₯ = 0.
3. The set S consist of all numbers of the form π‘Ž + 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
π‘₯ βˆ— 𝑦 = ln(𝑒 ; + 𝑒 < βˆ’ 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 π‘₯ βˆ— 𝑦 = (π‘₯ % + 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 π‘₯, 𝑦, π‘˜ ∈ 𝑅. An operation * is defined by π‘₯ βˆ— 𝑦 =
* is associative.
;<
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where π‘₯ + 𝑦 β‰  π‘˜. Show that
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