Every element a of some ring (R,+,0) satisfies the equation aoa = a.
Decide whether or not the ring is commutative.

Let F be the collection of all functions f: {1,2,3} → {1,2,3}. If f and g ∈ F, define an equivalence relation ~ by f ~ g if and only if f(3) = g(3).
a) Find the number of equivalence classes defined by ~.
b) Find the number of elements in each equivalence class.

Let f be a function defined by

Find the values for the constants a, b, c and d so that f is continuous and differentiable every where on the real line.

(a) Mr. X claims the following:
If a relation R is both symmetric and transitive, then R is reflexive. For this, Mr. X offers the following proof:
"From xRy, using symmetry we get yRy. Now because R is transitive, xRy and yRy together imply xRx. Therefore, R is reflexive."
Briefly point out the flaw in Mr. X's proof.

(b) Give an example of relation R which is symmetric and transitive but not reflexive.

Let X and Y denote the sets containing 2 and 20 distinct objects respectively and F denote the set of all possible functions defined from X to Y. Let f be randomly chosen from F. The probability of f being one-to-one is ______.