Can Basic Set Theory Language Help Understand Group Theory and Matrix Math?

[rebeka]

I'm reading the book Basic Set Theory by: Azriel Levy as I thought it might help me better understand Group Theory and Matrix Math. I have read the first chapter a number of times but I keep getting hung up on some of the syntax of the basic language or language of first-order predicate calculus with equality.

In the first two pages I get stuck on this:

i) $$\exists x\left ( x \in y \land \phi \right )$$

where this is read "there is an x in y such that phi!"

Such that phi what? I mean there are a lot of things I understand about what is being laid out in the language of first order logic such as:

ii) $$\phi \land \psi$$ is $$\lnot ( \lnot \phi \lor \psi )$$

but again I have trouble with i).

I really thought I would get somewhere with this book and still probably will over a lengthy period of time. Is there something else I should be addressing first? What does i) mean? Where are all these brackets coming from and where were they supposed to have been defined?

iii) $$R[A]=\left \{ y| \left ( \exists x \in A \right ) \left ( <x, y> \in R \right ) \right \}$$

There were lengthier examples! I think the brackets and the sudden realization of functions of the basic language are my two biggest hangups in being able to understand the full depth of the axioms being presented; many of which I have some vague understanding of from their general use in other subjects. Advice $$\land \lor$$ explanation of my above two dilemmas? :/

thx,
BekaD:

 

[Office_Shredder]

"such that phi" means "such that phi is true". It might seem weird to neglect the "is true" part, but when you think about actual statements like this it makes sense.

For example "There exists x in the real numbers such that x²=2". You wouldn't say There exists x in the real numbers such that x²=2 is true" (here phi is the statement $$x^2=2$$)

I don't know what the third part is supposed to be about, maybe some added context would be helpful? My first guess is that R is a relationship and <x,y> is just an ordered pair

 

[Hurkyl]

The typical application of the product defined in (iii) is to talk about the image of a function or relation.

If f is the function from the reals to itself defined pointwise by f(x) = x + 3, then we often encode f as its graph -- the set of all pairs (x,y) such that y = f(x).

So $f = \{ \langle x, x+3 \rangle \mid x \in \mathbb{R} \}$

If A is the set {1} and B is the set [0,1] (the interval of real numbers), then can you tell me what f[A] and f are?

 

[rebeka]

With respect to iii) it was an example where I am becoming confused with respect to the usage of brackets and what they signify. Different brackets seem to have different meanings at different times.

In this case the square brackets are addressing the class $$A$$ where the definition in the text for the use of the square brackets is to distinguish $$x$$ as being a subset or a () $$Dom(F)$$. I understand that these <> brackets denote ordered pairs but $$< x | x \in V >$$ leaves me asking what the exact significance of the use of the brackets chosen is ... I'm sure I'll pick it up it's just been so hard to learn the things I want to learn because I don't have a proper grasp of the generally accepted language used to describe.

Thanks for your replies and for the response to i). Both of your general explanations made things a little clearer for me. :)

 

[blue_raver22]

Hello BekaD,

I understand your confusion with the syntax and language of set theory. It can be quite complex and take some time to fully grasp. Let me try to provide some clarification on the concepts you mentioned.

First, let's start with the notation \exists x\left ( x \in y \land \phi \right ). This is a way to express the existence of an element x in a set y that satisfies a certain property, represented by the predicate \phi. So, when we read it as "there is an x in y such that phi!", it means that there exists at least one element x in the set y that satisfies the property \phi. The brackets are used to group the elements and operations in a logical expression.

Next, let's look at the notation \phi \land \psi is \lnot ( \lnot \phi \lor \psi ). This is known as De Morgan's Law, which states that the negation of a conjunction (AND) is equivalent to the disjunction (OR) of the negations of the individual statements. In simpler terms, it means that "not (A and B)" is the same as "(not A) or (not B)". This law is often used in logic and set theory to simplify expressions.

Finally, let's discuss the notation R[A]=\left \{ y| \left ( \exists x \in A \right ) \left ( <x, y> \in R \right ) \right \}. This is used to define the concept of a function in set theory. The notation R[A] refers to the set of all elements y for which there exists an element x in the set A such that the ordered pair <x,y> is an element of the relation R. In simpler terms, it means that R[A] is the set of all outputs y that can be obtained by inputting elements from A into the relation R.

I understand that these concepts can be confusing, but with time and practice, you will be able to fully understand and use them in your study of set theory. My advice would be to continue reading and practicing with examples, and don't hesitate to ask for clarification when needed. Best of luck in your studies!