Addition and multiplication in Z/nZ

In summary, the conversation discusses the concept of well-defined addition and multiplication in Z/nZ, where n is a modulus. It is important to check that these operations are well-defined because a single number in Z/nZ can represent an infinite set of numbers. The goal is to show that if two numbers a' and b' are in the sets a (mod n) and b (mod n), then their sum a' + b' is also in the set a + b (mod n). This involves finding an integer k such that a+b - (a'+b') = kn.
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I need to show that addition and multiplication are well-defined in Z/nZ. So far, I've figured out that I need to work within this framework:

Suppose a ~ a' and b ~ b'. Try to prove a + b ~ a' + b'.

If a ~ a'...something
If b ~ b'...something

Any ideas?
 
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  • #2
it sounds like you have a hazy notion of why we need to check that addition and multiplication are well-defined, and what it involves.

after all, setting

3 (mod 7) + 4 (mod 7) = 0 (mod 7)

makes perfect sense.

but here's the thing:

3 (mod 7) isn't just a single number: it's a whole slew of numbers

3 (mod 7) = {...,-11,-4,3,10,17,24,...}

so when we're using 3 (mod 7) in a sum, we might pick anyone of this infinite set as a "representative". so if we use -11, say, instead of 3, as a representative, we want to make sure that doesn't get us into trouble. so, even though:

3 (mod 7) looks like a single number, it's actually a stand-in, for an entire set.

now, what does it mean for a number k to be in the set 3 (mod 7)?

it means k = 3 + 7*something, or, as its usually put:

3-k is a multiple of 7. so, if

a ~ a', that is a = a' (mod n), what this MEANS is a-a' is a multiple of n, say sn.

b ~ b', so that b = b' (mod n), means b and b' differ by a multiple of n, b-b' = tn.

what you need to show, is if a' is in a (mod n), and b' is in b (mod n) (remember, these are sets), then a'+b' is in a+b (mod n), which is to say:

a+b - (a'+b') = kn, for some k (this integer k is what you are really trying to find. if you can find it, all is good. if you can't...houston, we have a problem).

now you KNOW a - a' = sn, and b - b' = tn. time to do a lil algebra now.
 

1. What is Z/nZ in terms of addition and multiplication?

Z/nZ refers to the set of integers modulo n, where n is a positive integer. Addition and multiplication in Z/nZ are operations that follow specific rules and properties, such as closure, commutativity, and associativity, which make them useful for solving equations and finding patterns in modular arithmetic.

2. How do you perform addition in Z/nZ?

To perform addition in Z/nZ, you first add the two integers as you would normally. Then, you divide the result by n and take the remainder as the final answer. This is known as the modulo operation, and it ensures that the result is always within the set of integers modulo n.

3. What is the difference between addition and multiplication in Z/nZ?

Addition and multiplication in Z/nZ are both binary operations, meaning they involve two operands. However, addition is commutative and associative, while multiplication is only associative. Additionally, the identity element in addition is 0, while the identity element in multiplication is 1.

4. How can you solve equations involving addition and multiplication in Z/nZ?

To solve equations in Z/nZ, you can use the same rules and properties that you would use in normal arithmetic, such as the distributive property and the inverse property. However, you must remember to apply the modulo operation after each step to ensure that the result stays within the set of integers modulo n.

5. What are some real-world applications of addition and multiplication in Z/nZ?

Addition and multiplication in Z/nZ have various real-world applications, including encryption, error detection in data transmission, and scheduling tasks in computer science. They are also used in fields such as cryptography, coding theory, and number theory to study patterns and structures in numbers.

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