Here is my 2 Questions
1) how do ns notate the set of irrational numbers?
2) just how do I display the over proof symbolically?
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You denote (not notate!) the collection of rational number by $sci-ch.orgbb Q$ and that that the genuine numbers through $sci-ch.orgbb R$ ; therefore the set of irrational numbers deserve to be composed as $sci-ch.orgbb R ackslash sci-ch.orgbb Q$ or $sci-ch.orgbb R - sci-ch.orgbb Q$, depending upon your taste. Her proof might simply go as adheres to : because $sqrt 2 in sci-ch.orgbb R ackslash sci-ch.orgbb Q$ however $(sqrt 2)^2 = 2 in sci-ch.orgbb Q$, $sci-ch.orgbb R ackslash sci-ch.orgbb Q$ is no closed under multiplication.
Hope the helps,
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Usually, the set of irrational numbers is written merely as $sci-ch.orgbb Rsetminus sci-ch.orgbb Q$.
As for the symbolic proof, my advice is "Avoid symbolic proofs."
A great proof in bsci-ch.orgematics is no one that is completely written with symbols alone. A an excellent proof is composed in words, yet is tho bsci-ch.orgematically rigorous, such the there is no doubt that any type of bsci-ch.orgematitian, if the really collection his mind to it, might write it under symbolically.
The proof, composed in bsci-ch.orgematics language, would certainly go choose this:
We wish to prove the $S=sci-ch.orgbb Rsetminus sci-ch.orgbb Q$ is not closed for multiplication. A collection is closed because that multiplication if:
$$forall x,yin S: xcdot yin S$$
This way the set is not closed for multiplication if$$exists x,yin S: xcdot y otin S$$
In our case, permit $x=y=sqrt 2$. Obviously, $x,yin S$, since $sqrt 2 otin sci-ch.orgbb Q$. Then, we deserve to see that
$$xcdot y = sqrt2 cdot sqrt 2 = 2 in sci-ch.orgbb Q,$$meaning the $xcdot y otin S$. This proves the $S$ is not closed for multiplication.