Download MATH 103: Contemporary Mathematics Logic Study Guide

MATH 103: Contemporary Mathematics
Logic Study Guide
1. Focus on the terminology discussed in the book and in class. You must know the meaning of the
following:
statement:
negation:
compound statement:
conjunction:
disjunction:
conditional:
antecedent:
consequent:
conclusion:
converse:
contrapositive:
inverse:
logical connectives:
logical argument:
2. Review the symbolic notation for negations and compound statements (examples on pages 19-20). You
should now know the meaning of the symbols ¬, ∨, ∧, →
3. If a statement is true, then its negation is
4. Review truth values and truth tables. Study the truth tables for p ∨ q, p ∧ q, p → q. You will find
them at page 22.
5. Explain in your own words why, if p is FALSE and q is FALSE, then the conditional statement p → q
is TRUE. (Remember the “promise not broken” examples we did in class).
6. You are given the three statements:
p: Old Grandpa feels good
q: He complains a lot
r: The weather is cold
(a) Write the following compound statement in symbolic form:
If the weather is cold, old Grandpa does not feel good and he complains a lot.
(b) Translate the following symbolic (compound) statement into English:
[(¬p) ∨ (¬q)] → (¬r)
(c) If p is false, q is true, and r is true, what is the truth value of [(¬p) ∨ (¬q)] → (¬r)?
7. Review when two statements are logically equivalent.
8. What is the negation of p → q? (Review this on p. 32.) Then write the negations of the following
statements
(a) If I am promoted then I will celebrate.
(b) If this child does not do her homework, I will not take her to the park.
9. Given the statements: p ∧ q, p ∧ (¬q), (¬p) → q, p ∨ q, ¬ [(¬p) ∨ (¬q)], are any two of them logically
equivalent? (Hint: use truth tables to answer this!)
10. Review the de Morgan’s Laws on negating conjunctions and disjunctions. What is ¬(p ∧ q) equivalent
to? And what is ¬(p ∨ q) equivalent to? Use truth tables to explain your answer.
11. Rephrase the following statements:
(a) It is not true that: the sun has not yet raised and the battle has begun.
(b) It is not true that: I am starving and, if you don’t give me some food then I’ll pass out.
12. Review the section on Tautologies and Contraditions, and practice problems 1, 2f on page 27-28.
13. Pay particular attention to the negation of statements containing quantifiers (i.e. referring to
groups of people, objects or animals). Review Section 2.8 p. 36-38.
14. What is the negation of “All men have big noses”? What about the negation of “Some people have
black hair” and of “No cat has green feet”?
15. Review very carefully the various forms of conditionals in English (Section 2.6 Conditionals in the
English Language and all problems in Exercise 2.6.1.) They can be confusing!
For example, do you know how to correctly rewrite the following statements in the form ”If . . . . . . ,
then . . . . . . ” ?:
(a) “For me to come, it is necessary I find some money.”
(b) “For me to come, it is sufficient I find some money.”
16. Given the conditional statement “I will warn you in advance if I cannot come to the meeting”
(a) Write it in symbolic form and identify the antecedent and the consequent.
(b) Form the converse.
(c) Form the contrapositive.
(d) Form the inverse.
17. Practice problems 2.7.2 no. 3b, d, e, h, j in Section 2.7.2.
18. Suppose a conditional statement is TRUE. Write T (True) or F (False) next to each of this statements.
Explain your reasoning.
(a) Its converse is true.
(b) Its converse is false.
(c) We cannot tell if the converse is true or false.
(d) Its contrapositive is true.
(e) Its contrapositive is false.
(f) We cannot tell if its contrapositive is true or false.
(g) Its inverse is true.
(h) Its inverse is false.
(i) We cannot tell the truth value of the inverse.
19. Venn (Euler) diagrams are used to determine the validity of “categorical syllogisms” (arguments involving quantifiers, i.e. words such as all, some, none, etc.). Read about these in Section 2.9.
20. Complete this definition:
“An argument is valid when, whenever ALL premises are true, then . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ”
21. Look at the examples discussed on pages 39-40: what is the distinctive feature of the Venn diagram
drawing in the case of an INVALID argument? The answer requires a bit of thought.
22. Is this argument containing quantifiers valid? Use Venn diagrams to justify your answer.
None of my pets is a dog. All dogs are cute. Therefore, all my pets are ugly.
23. Work through problems 2.9.2, 1c,e,g,i, and 2c.
24. Remember, even if the fact stated in the conclusion is false, that does not mean that the argument is
not valid! Give an example of a valid argument containing quantifiers, with a false conclusion. Show
its validity using Venn diagrams.
25. Use Venn’s diagrams to establish whether the following argument containing quantifiers is valid or not
valid.
All Zipfs are Poufs. All Poufs go to the beach. No Zonks go to the beach. Therefore, no Zonks are
Zipfs.
26. Review Section 2.10 on Analyzing Logical Arguments with Truth Tables logical arguments (remember that a logical argument is made of a list of premises and a conclusion) and write out the symbolic
form for each of the following arguments:
(a) Law of Detachment
(b) Law of Contraposition
(c) Disjunctive Syllogism
27. When is a logical argument valid? when is it invalid? (in this last case, it is called a fallacy). Explain
in your own words how you can use truth tables to determine the validity of an argument.
28. Practice problems 2.10.1 1a,d,e on page 48.
29. Write the symbolic form of the argument in problem 1c) on page 47. The form of this argument is
known as the Chain Rule. How many rows are there in the truth table that verifies the validity of
such an argument?
30. Warning: even if the fact stated in the conclusion of an argument is true, it does not mean that the
argument is valid. Give an example of a fallacious argument which has a true conclusion.
31. What is a fallacy?
32. Read about two common fallacies. Explain with an example and using truth tables why the Fallacy
of the Converse and the Fallacy of the Inverse are invalid arguments.
33. Do problems 2b, 3, 5 on pages 48-49.