CMSC 250 Homework 5 Fall 2001

Due Wed Oct 3 at the beginning of your discussion section.

If you do not write on your own paper, or if you write on both sides of your paper, or if you do not staple all of your homework answer papers together, you will lose points.

Use universal instantiation, universal modus tollens, or universal modus ponens to fill in valid conclusions for the following arguments:
1. If a real number x is a rational number, then it can be written as a ratio $\frac{a}{b}$ , where a and b are integers, and $b\neq 0$ .
  is a rational number.
  Therefore .
2. If n is an integer, and is odd, then can be written in the form for some integer k.
  5 is an integer, and $5^{2}=25$ is odd.
  Therefore, .
3. All CMSC classes are hard.
  CMSC250 is a CMSC class.
  Therefore, .
4. For all real numbers that are greater than 1, is a positive number.
  is not a positive number.
  Therefore, .
Decide if the following arguments are valid using universal modus ponens or universal modus tollens, or if they are invalid by the converse or inverse error.
1. All cars in good condition can be driven down the road.
  My car is not in good condition.
  Therefore, my car cannot be driven down the road.
2. All good computer programs contain comments that help the reader understand the code.
  My new computer program has comments that help the reader understand the code.
  Therefore, my new computer program is a good computer program.
3. All of your bases belong to us.
  The Alpha base does not belong to us.
  Therefore, the Alpha base is not one of your bases.
4. No new parent gets a lot of sleep.
  Sue is a new parent.
  Therefore, Sue does not get a lot of sleep.
Reorder the following premises and the final conclusion, replacing some premises with their contrapositives, to make it clear that the conclusion follows logically. You may add intermediate conclusions to help you follow the flow of the argument.
1. All citizens of Somerset Town over the age of 20 can run for the town council.
2. No one has moved to Somerset Town during the past year.
3. Bob lives in Somerset Town today, and moved there on his 24th birthday.
4. All people who live in Somerset Town at least a year automatically become citizens of the town.
5. Therefore, Bob can run for the town council.
Show whether the following arguments are valid or invalid. Support you answers by drawing Euler diagrams showing containment.
1. All dogs are animals.
  All dogs can bite people.
  Therefore, all animals can bite people.
2. All dogs are animals.
  All animals can bite people.
  Therefore, all dogs can bite people.
3. All airplanes have wings.
  No trucks are airplanes.
  Therefore, no trucks have wings.
4. All natural numbers are integers.
  All integers are rational numbers.
  All rational numbers are real numbers.
  All real numbers are complex numbers.
  Therefore, all natural numbers are complex numbers.
In each part, determine whether the following pairs of statements are equivalent or not.
1. $\exists x\in D$ , $P(x)\vee Q(x)\vee R(x) \equiv?\equiv$
  $\sim (\forall x\in D,\,\, \sim P(x)\wedge \sim Q(x)\wedge \sim R(x))$
2. $\sim (\exists p \exists b ((p\neq b) \wedge A(p,b))) \equiv ?\equiv \forall p\forall b ((p = b)\vee \sim A(c,s))$
Show that the following argument is valid by deducing the conclusion from the premises. Give justification for each step.
1. $\forall x\in D$ , $P(x)\rightarrow Q(x)$ .
2. $\forall x\in D,\,\, R(x)\vee P(x).$
3. $\forall x\in D,\,\, R(x)\rightarrow \sim S(x)$ .
4. $\exists y\in D,\,\, \sim Q(y)$ .
5. Therefore $\sim (\forall x\in D, \,\,S(x))$ .