Podcast
Questions and Answers
Is the sum of two irrational numbers always irrational? Provide a counterexample to support your answer.
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.
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, 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.
Is the statement 'All real numbers are rational numbers' true or false? Justify your answer.
Signup and view all the answers
If it is not true that 'All rectangle are squares', what can be concluded?
If it is not true that 'All rectangle are squares', what can be concluded?
Signup and view all the answers
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.
Studying That Suits You
Use AI to generate personalized quizzes and flashcards to suit your learning preferences.
Description
Test your understanding of real numbers and logical reasoning with these questions on irrational and rational numbers, and logical conclusions.
