Suppose x and y are irrational. Prove or disprove the following statement: “Either x + y or xy is irrational.”
Question
Suppose x and y are irrational. Prove or disprove the following statement:
"Either or is irrational."
Solution
This statement is not always true, and we can disprove it by providing a counterexample.
Consider the irrational numbers x = sqrt(2) and y = -sqrt(2). Both x and y are irrational numbers.
However, their sum x + y = sqrt(2) - sqrt(2) = 0, which is a rational number.
And their product xy = sqrt(2) * -sqrt(2) = -2, which is also a rational number.
So, in this case, neither x + y nor xy is irrational. Therefore, the statement “Either x + y or xy is irrational” is not always true.
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