Math 1002 Week 3 Mixed Problems

Week 3 Assignment 2: Mixed Problems

Please answer the following problems. It is required that you SHOW YOUR WORK to gain maximum points.

By Day 4, complete and submit your answers to the W3: Assignment 2 Dropbox.

On Day 5, post your answers and solutions for the assigned problems in the discussion thread as directed by your facilitator for review and comments from others. Do not post before Day 5.

Here are symbols you may need:      U ∩  [ ] COPY AND PASTE!

1. (4pts) Let p, q, and r be the following statements: p: It is raining. q: The clouds are dark. r: The temperature is dropping.

Translate the following statements into English (a) p  r It is raining and the temperature is dropping.

(b) ~p  (q  r) It is not raining and either the clouds are dark or the temperature is dropping

(c) q  r If the clouds are dark, the temperature is dropping.

(d) (  r   q)   p If either the temperature is not dropping or the clouds are not dark, then it is not raining.

2. (4pts) Write in symbolic form using p, q, r, , , , , where p, q, r represent the following statements:

p: A dog is friendly. q: A dog has a long tail. r: A dog licks faces.

(a) If a dog has a long tail, then it is not friendly. q  ~p

(b) If a dog is not friendly, then it does not lick faces. ~p  ~r

(c) If a dog has a long tail, then either the dog is friendly or the dog licks faces. q  (p v r)

(d) If a dog licks faces, then the dog is friendly and the dog has a long tail. r  (p ^ q) 3. (4 pts) Fill the headings of the following truth table using p, q, , , , and .

(a) (b)

p q (a) p q (b) T T F T T F T F T T F T F T F F T T F F F F F T

Part a: ~(p → q)

Part b: ~(p ^ q)

4. (4 pts) For each of the following conditionals, identify the antecedent and the consequent. Form the converse, inverse, and contrapositive.

(a) If I go to work, then I will get paid. Antecedent = I go to work Consequent = I will get paid Converse = If I get paid then I will go to work Inverse = If I don’t go to work, then I won’t get paid Contra = If I don’t get paid, then I won’t go to work

(b) Your yard will not look good if you don’t mow the grass.

Antecedent = You don’t mow the grass Consequent = Your yard will not look good Converse = If your yard doesn’t look good, you didn’t mow the grass Inverse = Your yard will look good if you mow the grass Contra = You will mow the grass if your yard looks good.

5. (2 pts) Rewrite ~ (p  ~q) using DeMorgan's laws.

Not (x or y) = not x and not y So: ~p ^ ~~q Meaning: ~p ^ q 6. (4 pts) (a) Translate the argument into symbolic form and (b) determine if the argument is valid or invalid. You may compare the argument to a standard form or use a truth table.

If it is raining, then we will close the window. We closed the window. ------ It is raining.

P = it is raining Q = close the window

Translation: p → q q therefore p

p q p → q

This is invalid, since the second bolded line has p false. 7. (4 pts) Use an Euler diagram to determine whether the syllogism is valid or invalid

All doctors are helpful. Some doctors are female. ------ Some helpful people are female.

This is VALID

8. (6 pts) Construct a truth table for ~q  (q  p).