Logic Homework

1. Are the following statements true or false? Explain/justify your answers.

a. The square of an odd number is odd.

b. All prime numbers are odd.

2. Are the following explanations/justifications true or false? Explain/justify your answers.

a. If one is healthy, then one is lucky. Tom is lucky. Therefore, Tom is healthy.

b. All joggers are happy. Some smokers are happy. Thus some smokers jog.

c. Only gourmets eat truffles. Some sows eat truffles. Therefore, some sows are gourmets.

d. All equilateral triangles are isosceles triangles. All isosceles triangles have an obtuse angle. Hence all equilateral triangles have an obtuse angle.

3. What conclusions, if any, can be drawn from the following statements? Explain.

a. Any friend of yours is a friend of mine. All of my friends are good-natured.

b. Some Wobbies are Looloos. Every Pimbolla is a Wobby.

c. If the decimal representation of a number repeats, then it represents a rational number. Pi is not a rational number.

d. If a nail is lost, then a shoe is lost. If a shoe is lost, then a horse is lost. If a horse is lost, then a rider is lost. If a rider is lost, then a battle is lost. If a battle is lost, then a kingdom is lost.

e. The Minnesota Vikings football coach tells his team before the last game of the regular season that if the team wins, they will be in the playoffs. The Vikings lost the game.

4. Prisoner Problem:

During an ancient war three prisoners were brought into a room. In the room was a large box containing five hats: two black and three white. Each man was blindfolded, and one of the hats was placed on his head. The men were then lined up, one behind the other, facing a wall. The blindfold of the man farthest from the wall was removed, and the man was permitted to see the hats on the heads of the two men in front of him. If he knew (not guessed) the color of the hat on his own head, he would be freed. He was unable to tell. Next the blindfold was taken off of the middle man. He could now see the color of the hat of the man in front of him but not behind him. This man had the same chance for freedom, but he also was unable to tell the color of his hat. The remaining man (still blindfolded) told the guards the color of his hat and he was released. What color hat was he wearing, and how did he know? Be sure your explanation is thorough.