Consequences
The Power Set Axiom allows a simple definition of the Cartesian product of two sets and :
Notice that
and thus the Cartesian product is a set since
One may define the Cartesian product of any finite collection of sets recursively:
Note that the existence of the Cartesian product can be proved without using the power set axiom, as in the case of the Kripke–Platek set theory.
Read more about this topic: Axiom Of Power Set
Famous quotes containing the word consequences:
“We are still barely conscious of how harmful it is to treat children in a degrading manner. Treating them with respect and recognizing the consequences of their being humiliated are by no means intellectual matters; otherwise, their importance would long since have been generally recognized.”
—Alice Miller (20th century)
“There is a delicate balance of putting yourself last and not being a doormat and thinking of yourself first and not coming off as selfish, arrogant, or bossy. We spend the majority of our lives attempting to perfect this balance. When we are successful, we have many close, healthy relationships. When we are unsuccessful, we suffer the natural consequences of damaged and sometimes broken relationships. Children are just beginning their journey on this important life lesson.”
—Cindy L. Teachey. “Building Lifelong Relationships—School Age Programs at Work,” Child Care Exchange (January 1994)
“Results are what you expect, and consequences are what you get.”
—schoolgirl’s definition, quoted in Ladies’ Home Journal (New York, Jan. 1942)