For each of the following relations, determine whether they are reflexive, symmetric, or transitive. Provide a reasoning.

3. The absolute difference of the integer numbers a and b is less than or equal to 3.
   • = {(a,b)|a,b Z |a b| 3}
4. The last digit of the decimal representation of the integer numbers a and b is the same.
   • = {(a,b)|a,b Z (a mod 10) = (b mod 10)}

Problem 3.3: total, injective, surjective, bijective functions

Are the following functions total, injective, surjective, or bijective? Expain why or why not.

1. f : N 7 N with f(x) = 2x²
2. f : R 7 R with f(x) = x² + 6

Problem 3.4: function composition

Given the functions f(x) = x + 1. g(x) = 2x, and h(x) = x², determine an expression for the following function compositions:

1. f g
2. f h
3. g f
4. g h
5. h f
6. h g
7. f (g h)
8. h (g f)

Problem 3.5: list comprehensions (haskell)

Your list comprehensions should be correct, they do not have to be efficient. You are not getting points for a list comprehension simply returning a hard coded solution list. In other words, your list comprehensions should continue to function correctly if parameters are changed.

210. Write a list comprehension that returns all positive factors of the number 210. Try to write the list comprehension in such a way that 210 can easily be replaced by a different number.
211. Write a list comprehension that returns a list of Pythagorean triads (a,b,c), where a,b,c are positive integers in the range 1..100 and the Pythagorean triad is defined as a² + b² = c². The list should not contain any duplicates where a and b are swapped. If the list contains (3,4,5) (since 3² + 4² = 25 = 5²), then is should not also include (4,3,5).