Writing and Proof: If true prove it, if false give a counterexample. Use contradiction when proving. (a) For each positive real number x, if x is irrational, then x^2 is irrational. (b) For every pair of real numbers a nd y, if x+y is irrational, then x if irrational or y is irrational

Chesley 2020-12-14 Answered
Writing and Proof: If true prove it, if false give a counterexample. Use contradiction when proving.
(a) For each positive real number x, if x is irrational, then x2 is irrational.
(b) For every pair of real numbers a nd y, if x+y is irrational, then x if irrational or y is irrational
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Clara Reese
Answered 2020-12-15 Author has 120 answers

(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 x2 is also irrational. For example:
consider x=5 ((it is positive real number) Then,
x2=(5)2
x2=5 {since, (a)2=a}
Since, 5 is not an irrational number because it can be written in the form of pq {where p and q are integer}.
That is 5=51.
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, x=ab {where a and b are the integers}
y=cd {where c and d are the integers}
According to the given statement:
x+y=irrational number
ab+cd= irrational number
ad+cdbd= irrational number
Let, ad+cd=p and bd=q.
Then,
pq= irrational number
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:
x=2,y=3, then x+y=2+3= irrational

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