. The relation ⋆ means “this number written out in English has this many letters”. For example, 6 ⋆ 3 because 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 has one copy of each element! (3) Is ⋆ reflexive, antisymmetric, symmetric, and/or transitive? Explain your answer.
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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 ⋆ 3
because 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 has
one 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 [your
language] 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 following
the arrows as far as you can. Where do you end up?
(8) Look at your digraph of ⋆ from question two. Imagine starting at 7 and following
the arrows as far as you can. Where do you end up?
(9) What do you think happens for ⋆ no matter what number you begin with, if you
keep following the arrows?
(10) Now look at your digraph of ♡. Does the same thing happen? Why or why not?
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Please solve from question 4 to 10. Thank you so much.
(4) Pick another language and let the relation ♡ mean “this number written out in [your
language] 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 following
the arrows as far as you can. Where do you end up?
(8) Look at your digraph of ⋆ from question two. Imagine starting at 7 and following
the arrows as far as you can. Where do you end up?
(9) What do you think happens for ⋆ no matter what number you begin with, if you
keep following the arrows?
(10) Now look at your digraph of ♡. Does the same thing happen? Why or why not?
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