Mathematical Systems!

[Arturia Pendragon]

Okay, in these problems we'll be devising our own mathematical systems. We can use whatever symbols we want and define the operation any way as well. We have to show an operation table and a way to describe the operation in words helps also.

On Number 1 it tells us to create a system with at least 5 symbols that does NOT have the closure property, but does have an identity element.

Number 2 tells us to create a system with at least 4 symbols that does not have the associative property.

And, lastly- Number 3 tells us to build an operation table for a 5-hour clock system with operation addition, and one with operation multiplication. It then asks which of the properties of mathematical systems does each system satisfy.

Please, at least point me to an area where I can learn all about this stuff! I am completely and utterly lost! ;-;

 

[JeffM]

It helps sometimes to give us a clue about what you are studying. Abstract algebra perhaps?

in this case, I at least do not know what you mean by "symbols." Are these symbols representing operations only, or do they pertain to elements and relations as well. For example, * is often used to stand for the operation of multiplication, = is often used to stand for the relation of equality, 10 is often used to refer to the tenth largest member of the set of positive integers, but it may also refer to the sixteenth largest member of that set.

 

[Arturia Pendragon]

It helps sometimes to give us a clue about what you are studying. Abstract algebra perhaps?

in this case, I at least do not know what you mean by "symbols." Are these symbols representing operations only, or do they pertain to elements and relations as well. For example, * is often used to stand for the operation of multiplication, = is often used to stand for the relation of equality, 10 is often used to refer to the tenth largest member of the set of positive integers, but it may also refer to the sixteenth largest member of that set.

I would assume that symbols stands for operations only, I don't exactly know. I have exactly what I posted and that's it. I'm sorry it's so little to go off of.

 

[JeffM]

Well here goes for number 1.

We have 1, 2, 3, 4, and \(\displaystyle \div\) interpreted in the normal way. We have 5 symbols. We have an identity element with respect to the operation of division with 1. We do not have closure because

\(\displaystyle 1 \div 3 \not \in \{1,\ 2,\ 3,\ 4\}\).

I suspect the exercise is more complex than this. What course are you studying?

 

[Arturia Pendragon]

Well here goes for number 1.

We have 1, 2, 3, 4, and \(\displaystyle \div\) interpreted in the normal way. We have 5 symbols. We have an identity element with respect to the operation of division with 1. We do not have closure because

\(\displaystyle 1 \div 3 \not \in \{1,\ 2,\ 3,\ 4\}\).

I suspect the exercise is more complex than this. What course are you studying?

The course is called the "Survey of Math." For #1 I need it to be a system with at least 5 symbols that does NOT have the closure property, but does have an identity element. I also need to show an operation table with my answer.

#2, it needs to be a system with at least 4 symbols that does not have the associative property. Also with an operation table.

and I don't even know if Question 3 has to do with 1 & 2.

 

[JeffM]

The course is called the "Survey of Math." For #1 I need it to be a system with at least 5 symbols that does NOT have the closure property, but does have an identity element. I also need to show an operation table with my answer.

#2, it needs to be a system with at least 4 symbols that does not have the associative property. Also with an operation table.

and I don't even know if Question 3 has to do with 1 & 2.

I apologize if this seems sarcastic, but there is no telling what a survey course may require. We can't be sure what the question means without any context from text or lectures. We have to guess. The simplest guess I can make is that the symbols are both operational symbols and object symbols. If that guess is correct, I gave you an answer for question 1. It has five symbols (one for an operation and four for the objects), an identity element, and lacks closure.

You need to ask your teacher or teaching assistant whether that answer is acceptable because I had to guess what the intent of the question was.