Which of the following statements are true? (a) x rational,...

Which of the following statements are true? (a) $x$ rational, $y$ irrational $\rightarrow x+y$ irrational. (b) $x$ rational, $y$ rational $\rightarrow x+y$ rational. (c) $x$ irrational, $y$ irrational $\Rightarrow x+y$ irrational. Prove the true ones and give a countercxample for each of the false ones.

Which of the following statements are true?
(a) $x$ rational, $y$ irrational $\rightarrow x+y$ irrational.
(b) $x$ rational, $y$ rational $\rightarrow x+y$ rational.
(c) $x$ irrational, $y$ irrational $\Rightarrow x+y$ irrational.
Prove the true ones and give a countercxample for each of the false ones.

Key Concepts

Counterexamples in Mathematical Proof

A counterexample is used to demonstrate that a general statement is not universally valid. By providing a specific example that contradicts a claim, one can show that the claim is false. This concept is crucial when evaluating statements that assert a property that may not hold under all circumstances.

Closure Properties in Arithmetic

Closure properties describe how sets of numbers behave under operations. For instance, the set of rational numbers is closed under addition, which is a key concept when evaluating expressions involving sums of rational numbers. However, the sum of certain irrational numbers can result in a rational number, highlighting that irrational numbers are not closed under addition.

Irrational Numbers

Irrational numbers cannot be expressed as the quotient of two integers. Their decimal representations neither terminate nor repeat. They exhibit different behavior in arithmetic operations compared to rational numbers, and adding irrational numbers can sometimes produce a rational result or remain irrational.

Rational Numbers

Rational numbers are numbers that can be expressed as the quotient of two integers, with the denominator not equal to zero. They have well-established algebraic properties, including closure under addition, meaning the sum of any two rational numbers is always rational.

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