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Question 1:

1. Fill in the blanks to rewrite the following statement:

For all objects Q, if Q is a quadrilateral then Q has four sides.

(a) All quadrilaterals ______________________________________.

(b) Every quadrilateral_____________________________________.

(c) If an object is a quadrilateral, then it___________________________.

(d) If Q is a quadrilateral, then Q__________________________.

(e) For all quadrilaterals Q,_________________________________.

2. (a) Is {4} ? {2, 4, 6, 8}?

(b) Is {3} ?Â {1,3,5,7}?

(c) Is {7} ?Â {{2},{5},{7}}? Â Careful.

(d) Is {8} ?Â {{2},{6},{8}}?

Â

3. Let A= {2,5,7} and B = {15,16,17,18} , and define a relation R from A to B as follows:

For all (x, y) ?Â A Ã— B , (x, y) ?Â R ?Â y/ x is an integer.

(a) Â Â Is 2 R 15? Â Â Is 2 R 16? Â Â Â Â Â Â Is (15, 5) ?Â R? Â Â Â Â Â Â Â Is (3, 18) ?Â R?

(b) Â Â Write R as a set of ordered pairs.

(c) Â Write the domain and co-domain of R.

(d)Â Draw an arrow diagram for R.

(e)Â Is R a function from A to B? Explain.

4. Define functions P and H from R to R by the following formulas:

P(x) = (x+ 1)(x?3) and H(x) = (x?2)^{2}Â ? 7.

Does F = G? Explain

**Â **5. Write a negation for each of the following statements:

(a) Â The variable P is defined and the lists are out of order.

(b) The variable P is defined or the lists are out of order.

(c)Â If Jack was with Joe on the fifteenth, then Jack is innocent.

(d) Â ?7 ? x < 3 (where x is a particular real number)**Â **

Â

6. Are the following statement forms logically equivalent : Â

p ? q ? p and Â Â p ? (?p ? q)?

Include a truth table and a few words explaining how the truth table supports your answer.

7. Write the following two statements in symbolic form and determine whether they are logically equivalent. Â Include a truth table and a few words explaining how the truth table supports your answer. Â Â Â

If Alice is out of Starbucks , then Alice is out of coffee.

Alice Â is not out of coffee or Alice Â is not out of Starbucks.

8. Write the converse, inverse, and contrapositive of

â€œIf Kevin is Tomâ€™s brother , then Mary Â is Tomâ€™s grandmother.â€

Original statement:

Converse:

Inverse

Contrapositive

9. Rewrite the following: Acceptance into the program is a necessary condition for graduating within the program.

Consider the argument form: Â

p Â ? ?q Â ? r

p ? q

q ? p Â Â Â

r.

Use a truth table below to determine whether this argument form is valid or invalid. Annotate the table (as appropriate) and include a few words explaining how the truth table supports your answer.

11. Write the form of the following argument. Â Â Is the argument valid or invalid?

Justify your Â answer.

If 673, 873 is a prime number, then 19 is not a divisor of 673,873. Â Â Â Â

19 Â Â is a divisor of 673,873. Â Â Â Â Â Â Â Therefore, 673,873 is not a prime number.

12. On the island of knights and knaves, you meet three natives, A, B, and C, who address you Â Â as follows:

A: At least one of us is a knave.

B: At most two of us are knaves. Â Â Â Â

What are A, B, and C? Â Â Explain. Â Â

13. Draw the circuit that corresponds to the following Boolean expression:

(P ? Q) ? (~P ? ~Q).

Do not simplify the circuit; just draw the one that exactly corresponds to the expression.

14. Find Â sum and difference ofÂ 10101_{2}and Â 1101_{2}.

Sum

Difference

Â

15. Write 101110_{2 }in decimal form.

Â

16. Write the 8-bit twoâ€™s complement for - 48.

Â

17. Consider the statement â€œThe cube of any odd integer is odd.â€

(a) Rewrite the statement in the form _

? ___________n Â , _____________

(Do not use the words â€œifâ€ or â€œthen.â€)

(b) Rewrite the statement in the form

?_________________ n Â ,

if_____________________ then_____________________ .

(Make sure you use the variable n when you ?ll in each of the second two blanks.)

(c) Write a negation for the statement.

18. Rewrite the following statement formally.

Use variables and include both quanti?ers ?Â Â Â and ? in your answer.

Every even integer greater than 2 can be written as a sum of two prime numbers.

19. Write negations for each of the following statements:

For all integers n, if n is even then n is not prime.

? real non-zero numbers x, if x < 1 then ^{1}/ _{x}Â > 1.

(c) For all integers a and b, Â Â if a^{2}Â divides b^{2}Â then a divides b.

20. Usingcfor â€œit is coldâ€ and w for â€œit is windyâ€, write

Â â€œToÂ beÂ windyÂ itÂ isÂ **necessary**Â thatÂ itÂ beÂ coldâ€Â inÂ symbols. Â

21. For each of the following statements, (1) write the statement informally without using

variables or the symbols ?Â or ?, and (2) indicate whether the statement is true or false and brie?y justify your answer.

(a) ? real numbers x, ? a real number y such that x < y.

(b) ? a real number y such that ? real numbers x, x < y.

22. In the questions below suppose the variable xrepresents students, S

F(x) means â€œxÂ is a freshmanâ€, and B(x) means â€œxÂ is an Biology Â majorâ€.

Â Match the statement in symbols with one of the English statements in this list:

- Some Freshman Â are Biology majors. Â
- Â Every Â Biology major is a freshman. Â
- No Biology major is a Freshman.
- $x Â (B(x) Â®~F(x)). Â Â Â Â Â Â Circle Â Â 1, Â Â 2, Â Â or Â Â 3
- Â Â "x (F(x) Â®~B(x)). Circle Â Â 1, Â Â 2, Â Â or Â Â 3
- Â ~$x Â (B(x) Ã™~F(x)). Circle Â Â 1, Â Â 2, Â Â or Â Â 3

Determine whether the following argument is valid.: Â Explain your reasoning

pÂ ?Â r

q ? r

q ? Â¬r

... Â¬p

Is the following argument valid or invalid? Justify your answer.

All prime numbers greater than 2 are odd.

The number a is not prime.

Therefore, the number a is not odd.

Show that the premises and determine if the argument is valid or invalid.

Every student in this class passed the first examâ€ and â€œBrian is a student in this classâ€ imply the conclusion â€œBrian passed the first examâ€.

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