R.3. Consider the relation R = {(m,n):m,n E N & m|n}. Prove that R defines a partial order on the set of natural numbers. Then show that R does not define a linear ordering on the set of natural numbers by giving two natural numbers a and b such that a does not divide b and b does not divide a.

A quadratic fuction f is given in photo provide

question 16

Please help me solve from question 4 to 10. Thank you.  For this Big Problem, we’re going to experiment with some unusual relations. The relation⋆ means “this number written out in English has this many letters”. For example, 6 ⋆ 3because the word “six” has 3 letters in it. (1) List the elements of ⋆ on {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}. (2) Draw a digraph of ⋆ on {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}. Remember that a digraph just hasone copy of each element! (3) Is ⋆ reflexive, antisymmetric, symmetric, and/or transitive? Explain your answer. (4) Pick another language and let the relation ♡ mean “this number written out in [yourlanguage] has this many letters”. List the elements of ♡ on {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}. (5) Draw a digraph of ♡ on {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}. (6) Is ♡ reflexive, antisymmetric, symmetric, and/or transitive? Explain your answer. (7) Look at your digraph of ⋆ from question two. Imagine starting at 10 and followingthe arrows as far as you can. Where do you end…