The TA said that we're supposed to approach the problem as to show 2Z + 5Z = {a+b | a in 2Z and b in 5Z} = Z. I forgot to paste that into #2. Other than that, you have 2Z and 5Z right. I'm sorry. I was in a rush to get to class and I didn't take my time typing out the post.Mark44 said:Maybe I'm looking at this the wrong way, but it doesn't seem true to me. I'm assuming that 2Z = {..., -4, -2, 0, 2, 4, 6, ...} and that 5Z = {..., -10, -5, 0, 5, 10, 15, ...}
I'm also assuming that + means "union."
If my assumptions are reasonable, every element of 2Z + 5Z is in Z, but there are a lot of elements in Z that aren't in 2Z + 5Z, such as 3, 7, 9, 11, 13, 17, and so on.
Abstract Algebra is a branch of mathematics that studies algebraic structures, such as groups, rings, and fields. It deals with the properties and structures of mathematical objects and their operations.
The notation "2Z + 5Z" represents the sum of two subsets of integers, where 2Z is the set of all even integers and 5Z is the set of all integers divisible by 5. This means that 2Z + 5Z is the set of all integers that can be written as a sum of an even integer and a multiple of 5.
To show that two sets are equal, we need to prove that they have the same elements. In this case, we can show that every element in 2Z + 5Z is also in Z and vice versa. This can be done by using the definition of 2Z + 5Z and Z and showing that they have the same elements.
Proving that 2Z + 5Z = Z is significant because it shows the relationship between different subsets of integers. It also demonstrates the closure property of addition, which means that the sum of two elements in a set will always result in an element within that same set.
Abstract Algebra has various applications in fields such as computer science, physics, and cryptography. It is used in coding theory to design efficient error-correcting codes, in physics to describe symmetries and transformations, and in cryptography to develop secure encryption algorithms.