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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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

Clara Reese · Accepted Answer

(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

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

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