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1.

We will say that two numbers between 1 and 31 are related, written

a \sim b

if their difference is a multiple of 7. So for example,

3 \sim 24,

since

24 - 3 = 3 \cdot 7,

but

3 ≁ 15

since

15 - 3 = 12

which is not a multiple of 7.

(a)

Which of the following are true? That is, which of the following pairs of numbers are related as we have defined above?

$4 \sim 14$
$7 \sim 14$
$10 \sim 17$
$17 \sim 24$
$10 \sim 24$
$20 \sim 10$
$31 \sim 3$
$25 \sim 25$

(b)

Which of the following statements are true about the

\sim

relation in this case?

$a \sim a$ for every number $a$
$a ≁ a$ for any number $a$
For any numbers $a$ and $b,$ if $a \sim b,$ then $b \sim a$
For any numbers $a$ and $b,$ if $a \sim b$ and $b \sim a,$ then $a = b$
For any numbers $a$ and $b,$ if $a \sim b$ and $b \sim c,$ then $a \sim c$

(c)

We will write

[a]

for the set of all numbers related to

a .

For example,

[7] = {7, 14, 21, 28} .

Find each of the following:

$[1] =$ ;
$[2] =$ ;
$[3] =$ ;
$[4] =$ ;
$[5] =$ ;
$[6] =$ .

Are there any numbers that are in more than one of the sets

[a]

above?

2.

When you divide a multiple of 7 by 7, you get a whole number. If you divide another number by 7, you can either write the result as a decimal or as a quotient and a remainder. For example,

19 \div 7

2

with a remainder of 5, since we can write

19 = 2 \cdot 7 + 5 .

The remainder is also called the modulus. When programming in python (and many other languages), the modulus operator is written as %. For example, 19 % 7 is 5. Try this out for a few numbers.

(a)

Find all the numbers

a

between 1 and 31 that are

5 \mod 7 .

That is, find all

a

such that a % 7 = 5.

(b)

Since the modulus is a function, each number has exactly one modulus when divided by 7. This means that the moduli partition the numbers from 1 to 31: every number belongs to exactly one of the sets of numbers with a particular modulus. We have already found the set for modulus 5. Find the other sets.

a % 7 = 0: ;
a % 7 = 1: ;
a % 7 = 2: ;
a % 7 = 3: ;
a % 7 = 4: ;
a % 7 = 6: .

(c)

We can use the moduli to define a relation on the numbers from 1 to 31. We will say that

a \sim b

if a % 7 = b % 7. In other words, two numbers are related if they belong to the same set of the partition we found above.

Which of the following are true? That is, which of the following pairs of numbers are related by this modulus relation?

$4 \sim 14$
$7 \sim 14$
$10 \sim 17$
$17 \sim 24$
$10 \sim 24$
$20 \sim 10$
$31 \sim 3$
$25 \sim 25$

Discrete Mathematics: An Open Introduction, 4th Edition

Worksheet Preview Activity
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Worksheet Preview Activity A4US

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Worksheet Preview Activity
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