Logic & Algebra: F Intersect G = ?

In summary: Group Theory by Samuel Eilenberg and Edward Freed. It is a comprehensive and thorough guide to group theory, with lots of examples and exercises.
  • #1
StephenPrivitera
363
0
All A's are B's.
can be written as
For all x, if x is A, then x is B.
If F = {x : x in domain, x is A}
and G = {y : y in domain, y is B}
Then I can write, "For all x, if x is A, then x is B" as
F intersect G = F

Similarly, I can write, "Some A's are B's" as
F intersect G [x=] [null]

I can write, "No A's are B's" as
F intersect G = [null]

I can write, "Only A's are B's" as
F intersect G = G

It seems that this approach might bring about considerale results (if only I knew more about the algebra of sets).
Is there some branch of logic that studies logic in this manner? Or is it simply more convenient to study logic conventionally? What is meant by the term "mathematical logic?"
 
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  • #2
Originally posted by StephenPrivitera
What is meant by the term "mathematical logic?"

I think that term refers to Symbolic Logic, which uses variables like p and q, and functions like "and", "if-then", and "if and only if", to form logical statements.
 
  • #3
The modern term for symbolic logic is propositional calculus. Sometimes more specifcally second order propositional calculus, the difference from first order being basically the quantifiers "There Exists" and "For All".
 
  • #4
Originally posted by selfAdjoint
The modern term for symbolic logic is propositional calculus. Sometimes more specifcally second order propositional calculus, the difference from first order being basically the quantifiers "There Exists" and "For All".

Interesting. I wasn't aware of this.
 
  • #5
A good google key is Zermelo-Frankel. This is the name of one of the systems of axioms for set theory, expressed mostly in the language of the propositional calculus. Also look up Foundations of Mathematics.

IIRC we had some threads about all this here in the old days.
 
  • #6
If you like logic and set theory, you might look into group theory (as well as rings and fields).

This is the book I have:
 

1. What is the meaning of "F intersect G" in logic and algebra?

The intersection of two sets, F and G, in logic and algebra refers to the elements that are common to both sets. In other words, it is the set of all elements that belong to both F and G.

2. How is "F intersect G" represented in mathematical notation?

The intersection of F and G is represented by the symbol ∩, so "F intersect G" can be written as F ∩ G.

3. What is the difference between "F intersect G" and "F union G"?

The intersection of F and G refers to the elements that are common to both sets, while the union of F and G refers to all the elements that belong to either F or G or both.

4. Can "F intersect G" be an empty set?

Yes, if F and G have no common elements, then their intersection will be an empty set. This means that there are no elements that belong to both F and G.

5. How is "F intersect G" used in computer science?

In computer science, the concept of intersection is commonly used in logic and set theory to determine relationships between sets of data. It is also used in programming languages to create conditions for data manipulation and filtering.

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