 # Discrete Mathematics Exam 1 Study Guide

1. Definitions:

Chapter 1: interrogatory, exclamatory, paradox, statement (proposition), truth value,
compound statement , logical connectives, truth table, conjunction , disjunction ,
implication (conditional) →, hypothesis (antecedent), conclusion (consequent), bicon-
ditional (equivalence) , negation ', well-formed formula (wff), tautology, absurdity
(contradiction), DeMorgan' s Laws , exclusive or , propositional calculus, argument,
premise (hypothesis), conclusion, valid, proof sequence, rules of equivalence , rules of
inference, quantifiers, universal quantifier , predicate, existential quantifier , scope,
free variable , predicate logic, rules of derivation

Chapter 2: theorem, inductive reasoning, deductive reasoning, counterexample, n fac-
torial n!, proof, proof by cases (exhaustion), direct proof, even, odd, contrapositive,
contradiction, rational number , perfect square, prime, composite, divides l , absolute
value, Principle of Mathematical Induction , basis step, inductive step, PMI (Principle of
Mathematical Induction), basis step , induction step, induction hypothesis, conclusion,
proof using PMI, recursive definition, sequence, term, recurrence relation, Fibonacci
sequence, initial condition, closed- form solution , summation notation , index of sum-
mation, first-order linear recurrence relation with constant coefficients, homogeneous,
particular solution, second-order, characteristic equation , characteristic roots

2. Determine whether a sentence is a statement

ex: Los Angeles is the most populous city in the U.S.

3. Determine whether a compound sentence is a compound statement

ex: Dr. Hein is not from Germany, or if he like dogs then he like pets

4. Write a compound statement as an English sentence

ex: If P: The sky is blue and Q: Jupiter had life on it 10,000 years ago, write P Q in
English

ex: With the same P and Q above, write (P Q) → (P') in English

5. Write an English sentence as a compound statement

ex: If I go to school today, I will ride my bike and not drive my car

6. Determine the truth value of a compound statement

ex: If P is false and Q is true, what is the truth value of (P Q) → (Q' P)?

7. From a given implication, form the converse, contrapositive and inverse.

ex: For R' → S, construct the converse, the contrapositive and the inverse

8. From a given implication, write the converse, contrapositive and inverse in an English
sentence

ex: For If I am late, then I drive fast, construct the converse, the contrapositive and
the inverse

9. Write equivalent variations of a conditional.

ex: Write "7 = 3 + 5 only if all dogs bark loudly" in an equivalent form.

10. Use the biconditional properly

ex: What is the truth value of R' S if R is true and S is false

11. Build truth tables of compound propositions (remember the standard order )

ex: Write the truth table for P' (Q' P)

12. Use propositional logic

ex: Show that [(P → Q) Q'] → P'

ex: Prove or disprove: 13. Quantify an open sentence

ex: "Every bat is blind"

ex: "Some dogs have spots"

14. Determine the truth value of a quantified open sentence. (Universe?)

ex: ex: ( y)[y has eight eyes]

15. Write a quantified statement as an English sentence.

ex: Consider the universe of math teachers. If P(x): x has big ears and Q(x): x writes
with chalk, write and in English

ex: Show that if x and y are even and odd integers (respectively), then y · (x+y) is an
odd integer

17. Prove by contraposition

ex: Show that if x is a rational number and y is an irrational number, then x + y is an
irrational number

18. Disprove by counterexample, etc.

ex: ex: 19. Prove/disprove "proofs" involving quantifiers

ex: There is a largest real number

20. Give an example to show that a certain deduction is false

ex: 21. Determine whether statements are equivalent

ex: Are P' (Q' P) and P Q equivalent?

22. Determine whether a propositional form is a tautology, absurdity or neither

ex: [P (P → Q)] Q'

23. Determine the validity of an argument

ex: I walk to work or I drive my car. If it rains, I do not walk to work. I drive my car.
Therefore, it rains.

24.Be able to use the PMI
ex: Show that for all natural numbers n
ex: Show that for all natural numbers n, 25. Find the value of a summation

ex: ex: 26. Find the first several terms of a sequence

ex: S(n) = 3n - (-1)n

ex: S(1) = -4; S(2) = -3; S(n) = S(n - 1) - S(n - 2)

27. Find a recursively defined sequence from an "nth-term formula"

ex: S(n) = 2n - 3

28. Find an nth-term formula for a recursively defined sequence (that is, solve a recurrence

ex: S(1) = -6; S(n) = 4 + S(n - 1)

ex: S(0) = 5; S(1) = 2; 2 · S(n) = S(n - 1) + S(n - 2)

• The examination should take about an hour.

• Part of the exam is "multiple choice", and part of it is "show your work". (You will
not need a ).

• The exam will be taken in the Testing Center [see additional instructions], and will
be open Thursday, September 24 and Friday, September 25.

• Please do all of your work on the white paper | the only thing that should be on the
examination paper itself is your name.

No cell-phone calculators will be allowed in the exam room!

• Good luck (if you are depending on luck)!

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