By A. N KhovanskiiÌ
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Extra resources for The application of continued fractions and their generalizations to problems in approximation theory
Consider the following statements: p: I throw a stone into a pond. q: Ripples are produced in the pond. If we write p → q, we are saying, If I toss a stone into a pond then ripples are produced in the pond. When we write p → q, we read it as If p, then q. Is this statement true? Our experience tells us that ripples are produced when a stone is tossed into a pond. The conclusion follows as a natural consequence of the premise that a stone is tossed into the pond. In mathematical terms, we are saying that the conclusion q follows as a result of p being true.
An idea closely related to a conditional statement p → q is that of its converse, which we write as q → p. We cannot conclude that the converse is true because the original statement is true. Consider the statement, If the traffic light is red then we stop the car. This statement is true for both legal and safety reasons. The converse is If we stop the car, then the light is red. This statement is not necessarily true; we can stop the car for a variety of reasons. ” When we write p ↔ q we recognize that the truth of either of the statements depends upon the truth of the other.
Determine which of the following statements are true. a. A positive integer is prime if and only if it is odd. b. An integer is divisible by 2 if and only if it is even. c. Figures are congruent if and only if they are similar. d. A four-sided figure is a parallelogram if and only if it is a rectangle. e. An integer is divisible by 5 if and only if it ends in a 5. f. An animal is a cat if and only if it has four legs. Application 4. a. State the converse of the Isosceles Triangle Property Theorem.