# Thread: Binary relation

1. ## Binary relation

Hello, i've been given this problem at University and I need some help with the formulas. From already thank you very much.

Let A = {1,2,3, ..., 365} be the set of the days of the year 2018, numbered from 1 (January 1, 1018) to 365 (December 31, 2018). Let R be the binary relation in A given by xRy if and only if x falls on the same day of the week as y.
a) How many equivalence classes are there? Write them, choosing a notation that you consider appropriate. Write the quotient set.
b) Write the relation R by a formula.
c) Given a day of the year x, write a formula that allows you to determine on what day of the week it falls. It will be useful to assign numbers to equivalence classes. It has the arithmetic operations and the functions "remainder" and "quotient", which take two integers a and b and return the remainder and the quotient, respectively, of the entire division of a by b. For example, remainder (17.3) = 2, quotient (17.3) = 5. (Data: on 1/1/2018 was Monday).
d) Write a formula that allows you to determine which day of the year (from 1 to 265) is a day of the given week (Monday, Tuesday, etc.) of week n of the year. Consider in this exercise that the week starts on Sunday. Verify that the formula works, consulting some days in a calendar.
e) Repeat the last two paragraphs for a year in which the 1st. January is Thursday.

2. ## Re: Binary relation

Originally Posted by FlippoLapi
Hello, i've been given this problem at University and I need some help with the formulas. From already thank you very much.

Let A = {1,2,3, ..., 365} be the set of the days of the year 2018, numbered from 1 (January 1, 1018) to 365 (December 31, 2018). Let R be the binary relation in A given by xRy if and only if x falls on the same day of the week as y.
a) How many equivalence classes are there? Write them, choosing a notation that you consider appropriate. Write the quotient set.
b) Write the relation R by a formula.
c) Given a day of the year x, write a formula that allows you to determine on what day of the week it falls. It will be useful to assign numbers to equivalence classes. It has the arithmetic operations and the functions "remainder" and "quotient", which take two integers a and b and return the remainder and the quotient, respectively, of the entire division of a by b. For example, remainder (17.3) = 2, quotient (17.3) = 5. (Data: on 1/1/2018 was Monday).
d) Write a formula that allows you to determine which day of the year (from 1 to 265) is a day of the given week (Monday, Tuesday, etc.) of week n of the year. Consider in this exercise that the week starts on Sunday. Verify that the formula works, consulting some days in a calendar.
e) Repeat the last two paragraphs for a year in which the 1st. January is Thursday.
Do you understand modulo arithmetic? That is required to answer this question.
Recall that $107 \equiv 2 \mod 7$ that means that the one hundredth seventh day of the year is on the same day of the week as the second day of January of that year.
Because there are only seven possible remainders, $\{0,1,2,3,4,5,6\}$, when an integer id divided by seven. a)Therefore there are seven equivalence classes.