I have a question in my discrete math class, groups question. Can anyone solve q

Question

I have a question in my discrete math class, groups question.

Can anyone solve question 1 for me with some explanation, thanks

Exercises 4.3 de which of the following sets are groups under the given operations: the set Q of rational numbers, under multiplication (i) the set of non-zero complex numbers, under multiplication; (ii) the non-zero integers, under multiplication (iv) the set of all functions from 11, 2, 3 to itself, under composition of functions; (v) the set of all real numbers of the form a b 2 where a and b are integers, under addition; (vi) the set of all 3 x 3 matrices of the form 1 a b 0 0 1 where a, b, c real numbers, under matrix multiplication; are the set of integers under subtraction; defined by the set of real numbers under the operation (vii) (viii) a b a b 2 f G Show that

Explanation / Answer

A group is a finite or infinite set of elements together with a binary operation (called the group operation) that together satisfy the four fundamental properties of 1.closure, 2.associativity, 3.The identity property, and 4.The inverse property

i) clearly the set of rational numbers follows closure property under multiplication

since the product of two rational numbers is always a rational numbers

in the similiar way we can prove that all the remainin properties also follows by rational numbers

like the identity element is 1

and inverse element of a is i/a provided a is not equal to zero.

but the set of rational numbers contains zero

it is failed to follow existence of inverse element

so it is not a group

ii) the set of complex numbers follows closure property under multiplication

it also follows all the remaining properties of group

it also exist inverse and identity for all elements , so it is a group

iii) the non zero integers under multiplication is also group since it follows all the 4 properties of group

so it is group

iv) the composition fo the function is closure, as well as asociative and identity also exist

inverse exist provided if it is one-one and onto but it is not possible for all function

so it need not be group

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