Generalize further and show that for any whole number r, the rth root r A is irrational

Which is the contrapositive of A B? (a) B A (b) B A (c) B A (d) A B 2. W

Which of the choices in Question 1 is the converse of A B?

Suppose that A B is true. Which is then automatically true, the converse or the contrapositive?

Restate as an implication: A triangle is a polygon.

In Exercises 38, state the negation.m and n are odd integers.

In Exercises 38, state the negation.Either m is odd or n is odd.

In Exercises 38, state the negation.x is a real number and y is an integer.

In Exercises 38, state the negation.f is a linear function.

In Exercises 914, state the contrapositive and converse. 9. If m and n are odd integers, then mn is odd.

In Exercises 914, state the contrapositive and converse.If today is Tuesday, then we are in Belgium.

In Exercises 914, state the contrapositive and converse.If today is Tuesday, then we are not in Belgium.

In Exercises 914, state the contrapositive and converse.If x > 4, then x2 > 16.

In Exercises 914, state the contrapositive and converse.

In Exercises 914, state the contrapositive and converse.

In Exercise 1518, give a counterexample to show that the converse of the statement is false. 15. If m is odd, then 2m + 1 is also odd.

If ABC is equilateral, then it is an isosceles triangle.

If m is divisible by 9 and 4, then m is divisible by 12.

If m is odd, then m3 m is divisible by 3.

In Exercise 1922, determine whether the converse of the statement is false. 19. If x > 4 and y > 4, then x + y > 8.

If x > 4, then x2 > 16.

If |x| > 4, then x2 > 16.

If m and n are even, then mn is even.

In Exercises 23 and 24, state the contrapositive and converse (it is not necessary to know what these statements mean). 23. If f and g are differentiable, then fg is differentiable.

If the force field is radial and decreases as the inverse square of the distance, then all closed orbits are ellipses.

In Exercises 2528, the inverse of A B is the implication A B. 25. Which of the following is the inverse of the implication, If she jumped in the lake, then she got wet? (a) If she did not get wet, then she did not jump in the lake. (b) If she did not jump in the lake, then she did not get wet. Is the inverse true?

State the inverses of these implications: (a) If X is a mouse, then X is a rodent. (b) If you sleep late, you will miss class. (c) If a star revolves around the sun, then its a planet.

Explain why the inverse is equivalent to the converse.

State the inverse of the Pythagorean Theorem. Is it true?

Theorem 1 in Section 2.4 states the following: If f and g are continuous functions, then f + g is continuous. Does it follow logically that if f and g are not continuous, then f + g is not continuous?

Write out a proof by contradiction for this fact: There is no smallest positive rational number. Base your proof on the fact that if r > 0, then 0 < r/2 < r.

Use proof by contradiction to prove that if x + y > 2, then x > 1 or y > 1 (or both).

In Exercises 3235, use proof by contradiction to show that the number is irrational. 32. 1 2

In Exercises 3235, use proof by contradiction to show that the number is irrational.3

In Exercises 3235, use proof by contradiction to show that the number is irrational.3 2

In Exercises 3235, use proof by contradiction to show that the number is irrational.4 11

An isosceles triangle is a triangle with two equal sides. The following theorem holds: If  is a triangle with two equal angles, then  is an isosceles triangle. (a) What is the hypothesis? (b) Show by providing a counterexample that the hypothesis is necessary. (c) What is the contrapositive? (d) What is the converse? Is it true?

Consider the following theorem: Let f be a quadratic polynomial with a positive leading coefficient. Then f has a minimum value. (a) What are the hypotheses? (b) What is the contrapositive? (c) What is the converse? Is it true?

Let a, b, and c be the sides of a triangle and let be the angle opposite c. Use the Law of Cosines (Theorem 1 in Section 1.4) to prove the converse of the Pythagorean Theorem.

Carry out the details of the following proof by contradiction that 2 is irrational (this proof is due to R. Palais). If 2 is rational, then n 2 is a whole number for some whole number n. Let n be the smallest such whole number and let m = n 2 n. (a) Prove that m

Generalize the argument of Exercise 39 to prove that A is irrational if A is a whole number but not a perfect square. Hint: Choose n as before and let m = n A n A , where x is the greatest integer function.

Generalize further and show that for any whole number r, the rth root r A is irrational unless A is an rth power. Hint: Let x = r A. Show that if x is rational, then we may choose a smallest whole number n such that nxj is a whole number for j = 1,...,r 1. Then consider m = nx n[x] as before.

Generalize further and show that for any whole number r, the rth root r A is irrational unless A is an rth power. Hint: Let x = r A. Show that if x is rational, then we may choose a smallest whole number n such that nxj is a whole number for j = 1,...,r 1. Then consider m = nx n[x] as before.