Question

Is it TRUE or FALSE ?

If a∈Q and b∉Q,then ab∉Q.

Prove your choice.

Answer 1

false if b ='s ∉ then a*b can't =∉

Answer 2

s means

Answer 3

\(a = p/q\)

and say \(ab \in \mathbb Q\), then :
\(ab = p/q*b = u/v\)
\(\implies b = (uq)/(vp)\)
contradiction - QED.

Answer 4

Can you give me some example?

Answer 5

example :
\(a = 2/3,~ b = \sqrt{2}\)

Answer 6

multiply them :
\(ab = 2/3 * \sqrt{2} \not \in \mathbb Q\)

Answer 7

\(\text{rational} \times \text{irrational} = \text{irrational}\)

Answer 8

so this statement is true

Answer 9

Yep !

Answer 10

but in answer this statement is false

Answer 11

this is a trivial proof, so i wont argue :
http://math.stackexchange.com/questions/45104/prove-that-the-product-of-a-rational-and-irrational-number-is-irrational

Answer 12

wait a sec, the given statement is FALSE when \(a = 0\)
in all other cases it is TRUE

Answer 13

so your textbook answer is right :
the counter example is : \(a = 0, ~~b = \sqrt{2}\)
\(ab = 0*\sqrt{2} = 0 \in \mathbb Q\)

Answer 14

yeah....thank you so much

Answer 15

you're welcome !