English | Español

Try our Free Online Math Solver!

Online Math Solver

 

 

 

 

 

 

 

 
 
 
 
 
 
 
 
 

 

 

 
 
 
 
 
 
 
 
 

Please use this form if you would like
to have this math solver on your website,
free of charge.


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

16. Prove by contradiction

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
relation); also, check your answer(s)

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)

Notes about the examination:

• 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)!

Prev Next