4 Are the following statements true or false If they are tru

(4) Are the following statements true or false? If they are true, prove them, and if they are false provide a counterexample. (a) If r set membership Q and x not set membership Q,then rx not set membership Q. (b) The product of two irrational numbers is irrational. (e) For all real numbers x, root x set membership Q.

Solution

(a)

TRUE

proof:

let rx belongs to Q => rx = a/b for some integers a,b

r belongs to Q => r = c/d for some integers c,d

=>

x = (ad)/(bc) => x belongs to Q (since ad, bc are also integers) which is a contradiction to the given hypothesis

=>

rx doesnt belong to Q

thus proved

(b)

False

counter example :

sqrt(2) * sqrt(2) = 2

(c)

False

counter example

sqrt(2) doesnt belong to Q