A hopeful irrational number \$\$q\$\$ is by an interpretation a actual number 보다 cannot be expressed as a ratio of \$2\$ integers.To show that the set of irrational number is not closed under plain multiplication, I seek a counter-example the is \$\$sqrt2 imes sqrt2 = 2 = frac21\$\$which is obvious as can be viewed that the product of \$2\$ irrational number is a positive rational number i m sorry is not in the collection of positive irrational number.

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?

You are watching: Are irrational numbers closed under addition  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,

See more: What Does Shishi Mean In Chinese, What Does Shi Shi In Chinese Mean 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.