(a) For each positive real number x, if x is irrational, then
(b) For every pair of real numbers a nd y, if x+y is irrational, then x if irrational or y is irrational
(a) The given statement is false. Because for each positive real number x, if x is irrational number, then it is not necessary to their square
consider
Since, 5 is not an irrational number because it can be written in the form of
That is
Hence, given statement is false.
(b)The given statement is true.To prove this, contrary assume that x and y both are rational number. So,
According to the given statement:
x+y=irrational number
Let, ad+cd=p and bd=q.
Then,
rational number=irrational number
Which is the contradiction. Therefore, our consideration values {x and y both are rational number} are wrong. Hence, the given statement is true.
For example: