Write a program that determines whether a given relation on a set exhibits the following properties: reflexive, symmetric, transitive, and equivalence $($if it satisfies all three properties$).$ Please review section $4\mathrm{.}4$ of the textbook.

Input:

Take a list of pairs as input, representing a relation on a set. The elements in the pairs represent elements of the set on which the relation is defined.

Output:

Determine if the relation is:

i$.$ Reflexive: For every element a in the set, $(a,a)$ should be in the relation.

ii$.$ Symmetric: If $(a,b)$ is in the relation, then $(b,a)$ should also be in the relation.

iii. Transitive: If $(a,b)$ and $(b,c)$ are in the relation, then $(a,c)$ should also be in the relation.

iv$.$ Equivalence: Satisfied if the relation is reflexive, symmetric, and transitive.

Validate using the following test cases whether each property is satisfied for the relation.

Example test cases:

Test case $1$:

Input: $\{(1,1),(2,2),(1,2),(2,1)\}$

Output: Reflexive, Symmetric, Transitive, Equivalence

Test case $2$:

Input: $\{(1,1),(2,2),(1,2)\}$

Output: Reflexive, Not Symmetric, Transitive, Not Equivalence

Test case $3$

Input: $\{(1,2),(2,3)\}$

Output: Not Reflexive, Not Symmetric, Not Transitive, Not Equivalence.