sums from relations and functions

These are some sums from relations and functions class 12. Please help me by doing the sums in details with all statements written :

1. Let be the relation on ={1,2,3}given by ={(1,1),(1,2),(2,3),(3,2)}. Determine whether it is reflexive. Write down the smallest collection of ordered pairs to be added to to make it transitive.

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2. Consider the set =ℤ∩[−2,2). Write down the number of

a. relations on .

b.reflexive relations on .

3. Give examples (without proof) of

a. a function :ℝ→ℝsuch that is injective but not surjective.

b. a function :ℝ→ℝsuch that is surjective but not injective.

4. Determine whether the function :ℝ→(0,∞) given by ( )= ^2+1 is onto. Justify your answer.

5. Let and be two sets such that | |=3and | |=4. Write downa.the number of injective functions from to .b.the number of surjective functions from to .

6. Let be a relation on ℤ given by ={( , ): , ∈ℤ,2 +3 is divisible by 5}. Prove that is an equivalence relation. Find the equivalence class of the element 2∈ℤ.

7. Determine whether the function :ℝ→ℝ given by ( )= x^3/(x^2+1)  is bijective or not. Justify your answer.