Insights Blog
-- Browse All Articles --
Physics Articles
Physics Tutorials
Physics Guides
Physics FAQ
Math Articles
Math Tutorials
Math Guides
Math FAQ
Education Articles
Education Guides
Bio/Chem Articles
Technology Guides
Computer Science Tutorials
Forums
General Math
Calculus
Differential Equations
Topology and Analysis
Linear and Abstract Algebra
Differential Geometry
Set Theory, Logic, Probability, Statistics
MATLAB, Maple, Mathematica, LaTeX
Trending
Featured Threads
Log in
Register
What's new
Search
Search
Search titles only
By:
General Math
Calculus
Differential Equations
Topology and Analysis
Linear and Abstract Algebra
Differential Geometry
Set Theory, Logic, Probability, Statistics
MATLAB, Maple, Mathematica, LaTeX
Menu
Log in
Register
Navigation
More options
Contact us
Close Menu
JavaScript is disabled. For a better experience, please enable JavaScript in your browser before proceeding.
You are using an out of date browser. It may not display this or other websites correctly.
You should upgrade or use an
alternative browser
.
Forums
Mathematics
Set Theory, Logic, Probability, Statistics
How to interpret this definition of a subset?
Reply to thread
Message
[QUOTE="Stephen Tashi, post: 6054284, member: 186655"] The definition: ##A \subset B \iff \forall x ( x \in A \implies x \in B) ## doesn't have a problem when ##A## and ##B## are disjoint sets. In that situation, we have two possibilities - either there exists an element ##x## such that ##x \in A## or no such element exists. If there exists an ##x \in A## then there exists an ##x## such that the implication ##x \in A \implies x \in B## is false in a non-vacuous way. Hence the universally quantified statement ##\forall x (x \in A \implies x \in B) ## is false because there exists at least one ##x## that makes the implication false. It there does not exist an ##x \in A## then ##A = \emptyset## and the definition says ##A = \emptyset \subset B##. ( It's a tradition in mathematical writing to treat any variable variable that comes up as universally quantified unless specific restrictions are placed on it - a potentially confusing tradition. ) That's notation, but what would you mean by that notation? The fact that notation doesn't contain the symbol "##\implies##" doesn't guarantee that a coherent interpretation of the notation can omit the concept of "if...then..". [/QUOTE]
Insert quotes…
Post reply
Forums
Mathematics
Set Theory, Logic, Probability, Statistics
How to interpret this definition of a subset?
Back
Top