Discrete Math Exercises If a proof is given then label your variables, use proper algebra, and state a conclusion 2. Give P=1, Q=0, and R = 1what is S? 3.
Discrete Math Exercises If a proof is given then label your variables, use proper algebra, and state a conclusion 2. Give P=1, Q=0, and R = 1what is S?
3. Write the negation and contrapositive for the following statement.
?x, y ? R , if xy= 1 then one of the numbers is a reciprocal of the other.
Negation: ___________________________________________________________________
Inverse:___________________________________________________________________
Converse:________________________________________________________________
Contrapositive: ____________________________________________________________
5. Indicate whether the argument is valid or invalid. If valid then state which argument was used? If
invalid then state which error was used?
A)
If compilation of a computer program produces error messages, the program is not correct.
Compilation of this program does produce error messages.
Therefore, this program is not correct.
B) If the student is a freshman, then they must take writing.
Caroline is not a freshman.
Therefore, Caroline is not taking writing.
4. A set of premises and a conclusion is given. Use the valid argument forms to deduce the
conclusion.
a)
q ? r b) p ? q
c) ~ r
6. Let P(x) be the predicate x =
d) ?~ q ? ~ p
1
.
x
a) Write P(2), P(1/2), P(-1), and P(-1/2) and determine the truth value of the statement.
b) What is the domain of this P(x) if the xs belong to all Real Numbers?
7. a. Represent the decimal integer 329 in binary notation.
b. Represent 1001112 in decimal notation.
8. Determine whether the statement is true or false. Justify your answer with a direct proof or
contradiction proof or a counterexample.
If a is even integer and b is odd integer, then 3a2+ 2b2 is even.
9. Determine whether the following statements are logically equivalent by Supply a reason for
each step.
Prove: ~ (p ? ~q) ? (~ p ? ~q) ? ~p
~ (p ? ~q) ? (~ p ? ~q)
? (~ p ? q) ? (~ p ? ~q)
by __________________
? ~ p ? ( q ? ~q)
by_______________________
?~p ?(t)
by_______________________
?~p
by _______________________
10. Which of the following sets are equal and why?
A = {-1, -2,-3}
B = {?? ? ?| ? 3 ? ?? < 0}
C= {?? ? ?| ? 3 < ?? < 0}
E= {?? ? ?+ | ? 3 < ?? < 0}
D= {?? ? ?| ? 4 < ?? < 0}
11. Let C = D = {?? ? ?| ? 4 ? ?? < 2} and define a relation S from C to D as follows: For all
( x, y ) ? C ? D, ( x, y ) ? S , means that
1 1
? is an integer. Solve for the following.
x y
a. Is 2S5? ________ b. Is -1 S 1?________
c. Is 3 S 3?________
d. Write S as a set of ordered pairs._____________________________
e. Write the domain and co-domain.
Domain:______________________
Codomain:_____________________
f. Draw an arrow diagram for S.
12. Determine whether the statement is true or false. Justify your answer with a direct proof or
contradiction proof or a counterexample. There is no greatest integer
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