Problem 9. Determine if relations below are relations of equivalence or
not. If it is a relation of equivalence, then formulate what three properties
(reflexive, symmetric, transitive) mean in this case and prove them (if it is
not obvious). If not, explain what property does not work (give an example).
a) We say that sets A and B are equivalent iff $A \subset B$.
b) We say that two cities are equivalent if you can drive from one city to the
other only using roads that are two way roads.
c) We say that a city A is equivalent to a city B, if A = B or both of them
have airports and there was at least one plane in 2018 that flew from the
city A directly to the city B.
d) Student A is equivalent to a student B if they took at least one class
together.
e) A number $x$ is equivalent to a number $y$ if $xy \ge 0$.
f) A number $x$ is equivalent to a number $y$ if $x^2 \equiv y^2 \mod 10$.
