Podcast Beta
Questions and Answers
Is the sum of two irrational numbers always irrational? Provide a counterexample to support your answer.
No, the sum of two irrational numbers is not always irrational. A counterexample is √2 + (-√2) = 0, which is a rational number.
Prove that the product of two rational numbers is always rational.
Let a and b be two rational numbers. Then, a = p/q and b = r/s, where p, q, r, and s are integers and q, s ≠0. The product ab = (p/q) × (r/s) = (pr)/(qs), which is a rational number.
If p → q is true, and q is false, what can be concluded about p?
If p → q is true, and q is false, then p must be false. This is because if p were true, then p → q would imply that q is true, which contradicts the given fact that q is false.
Is the statement 'All real numbers are rational numbers' true or false? Justify your answer.
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If it is not true that 'All rectangle are squares', what can be concluded?
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Study Notes
Irrational and Rational Numbers
- The sum of two irrational numbers is not always irrational.
- A counterexample to this statement is the sum of √2 and -√2, which equals 0, a rational number.
Rational Number Product
- The product of two rational numbers is always rational.
- This is because the product of two fractions with integer numerators and denominators will always result in another fraction with integer numerators and denominators.
Conditional Statements
- If p → q is true, and q is false, then p must also be false.
- This is because the conditional statement p → q implies that if p is true, then q must also be true; if q is false, then p cannot be true.
Real and Rational Numbers
- The statement 'All real numbers are rational numbers' is false.
- This is because there are real numbers that are not rational, such as π and e, which cannot be expressed as fractions.
Rectangle and Square
- If it is not true that 'All rectangles are squares', then it can be concluded that there exist rectangles that are not squares.
- This is because the statement implies that not all rectangles have the additional property of having all sides of equal length, which defines a square.
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Description
Test your understanding of real numbers and logical reasoning with these questions on irrational and rational numbers, and logical conclusions.
