Homework 5

DueThursday, October 13, 2016

  1. The fallacy of “denying the antecedent” can be understood by way of an example of a faulty argument. Here is one: a tennis coach says, “If it rains tomorrow, the match will be canceled.” Then it turns out that it does not rain. However, on the way to the tennis match, the team’s bus breaks down. Therefore, the match is canceled. Did the tennis coach lie, or was his statement still factually true? (In every situation in which it were to rain, the match would have been canceled.) Come up with an example scenario for the fallacy of “affirming the consequent” (the fallacy which, given “p -> q” and “q”, seeks to conclude “p”): try to find a scenario which would help younger, elementary school-aged students, understand why this line of reasoning does not work.
  2. Suppose a, b, and c are natural numbers with a non-zero. Recall, “a | b” means “a divides b”, or, in other words, “a is a factor of b”. Find a counterexample to the statement “If a | bc, then a | b or a | c.” (That is, find natural numbers a, b, and c, such that a is a factor of the product bc, but a is not a factor of either b or of c).
  3. Suppose p is a prime number, b and c are any natural numbers, p | bc, and p is not a factor of b. By using the Fundamental Theorem of Arithmetic, explain (in words) why p must be a factor of c. (See how this question relates to the previous one? This means, in the counterexample you look for in (2), the number a cannot be prime).
  4.  In class, we drew a visual model for LCM(a, b) where a and b are natural numbers. As an example, if we are looking for the LCM of 6 and 10, we can draw lines of length 6 units and then, below those lines, draw  lines of length 10 units, until we get the same length (30 for both). Similar to this kind of visual model for LCM, come up with a visual model for GCF. (As a hint: it may help to assign a variable d for the GCF(a, b), and then draw a picture illustrating how d relates to a and b).
  5. Use the Sieve of Eratosthenes to find all prime numbers below 100.