HW 2: Well Ordering Principle and Intro to Propositions

 

Introduction

In this homework you will gain practice using the Well Ordering Principle in proofs and will begin to work with propositions and truth tables.

Exercises

Your solutions to these exercises should be typed up in LaTeX.
  1. Problem 2.4 from the text.

  2. Problem 2.7 from the text.

  3. Let P, Q, and R be the propositions:
    P: Grizzly bears have been seen in the area.
    Q: Hiking is safe on the trail.
    R: Berries are ripe along the trail.
    Translate the following assertions into propositional formulas using P, Q, and R and the propostional connectives AND, OR, NOT, IMPLIES, IFF.
    1. Berries are ripe along the trail, but grizzly bears have not been seen in the area.
    2. Grizzly bears have not been seen in the area and hiking on the trail is safe, but berries are ripe along the trail.
    3. If berries are ripe along the trail, hiking is safe if and only if grizzly bears have not been seen in the area.
    4. It is not safe to hike on the trail, but grizzly bears have not been seen in the area and the berries along the trail are ripe.
    5. For hiking on the trail to be safe, it is necessary but not sufficient that berries not be ripe along the trail and for grizzly bears not to have been seen in the area.
    6. Hiking is not safe on the trail whenever grizzly bears have been seen in the area and berries are ripe along the trail.

    1. Use truth tables to show that P IMPLIES Q is equivalent to NOT(P) OR Q. (This shows every implication can be written as an OR statement.)
    2. Use truth tables to show that P OR (Q AND R) is equivalent to (P OR Q) AND (P OR R). (This shows the rule for distributing OR over AND.)

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