please any one solve the questions in brief.

Answers
Step-by-step explanation:
Note that i will use for the relation symbol instead of the cursive S.
a)
Reflexive: is divisible by 5. As zero is divisible by any integer then the relation is reflexive.
Symmetric: We need that . If
then
, where c is divisible by 5. Then
. As c is divisble by 5, then -c must also be divislbe by 5. Hence, the relation is symmetric.
Transitive: We need that . By modular arithmetic we have that
and
, as
and
. Adding the equations yield
, which simplfies to
. Hence a - c is divisible by 5 and the relation is therefore transitive.
b)
Reflexive: Trivially, all lines lie in the same plane as it self. It is reflexive.
Symmetric: If a line L lies in the same plane as M, then M must lie in the same plane as L. It is symmetric.
Transitive: If two lines lie in a plane, , and we also have
, then
L, M and P are all in the same plane, or we can create a third plane that contains both L and P. Hence, it is transitive.
c)
Reflexive: Trivially, any number is a divisior of it self.
Symmetric: If and
, then a divides b. However, as
, then b cannot divide a (this would yield a fraction/decimal). Not symmetric.
Transitive: If , then
, for some number p. As
then for soome q. Then by substitution we have
. Hence c is a divisor of a, and we have that
. It is transitive.