Question 1088149
.
<pre>
Let me to educate you a bit (a little).


        The equality ad = bc means that the determinant of the matrix {{{(matrix(2,2, a,b, c,d))}}} is equal to zero.        (*)

        This equality also means that the vectors (a,b) and (c,d) are proportional.                  (**)


    These  (*)  and  (**)  are <U>EQUIVALENT properties</U>.


Having this HINT will help you easily answer your questions.


a)  <U>Regarding "reflexivity" property</U>

    The property (the term) "reflexive" means that element E satisfies the relation ErE.

    <U>ANY</U> element of your list satisfies this property.

        By the way, to check whether this property is valid, you <U>DO NOT NEED</U> to treat two different elements of your set.

        The reflexive property must be checked for EACH element individually, not for the pair of elements.

        In this sense the formulation a) in your post IS INCORRECT.


b)  <U>Regarding "symmetric" property</U>

    The example of two elements that demonstrate the symmetric property is 

    (1,2)  and  (2,4).

    You may check it immediately, making the necessary calculations manually.


c)  <U>Regarding "transitivity" property</U>.

    The property (**) makes transitivity OBVIOUS in this case:

         If vector A is proportional to vector B, And B is proportional to C, then CLEARLY A is proportional to C 

         and the transitivity is on the place.


d)  <U>Regarding classes of equivalency</U>.


    The element (1,2) has equivalent (2,4), (3,6).                  (1)

    The element (1,4) has equivalent (2,8), (3,12).                 (2)

    The element (1,3) has equivalent (2,6).                         (3)

    The element (3,4) is equivalent to itself and to nothing else.  (4)


    So, the lists (1), (2), (3) and (4) represent four classes of equivalency.
</pre>

Solved.