Discrete Mathematics Answers

given x = [101011110]  and y = [110101001] find xoy and x ^ y

Use Deductive reasoning to solve the following logic puzzle.

1. Four kids went to an unusual pet store. Each child picked out a different animal to 

take home. Can you match the child with their new friend?

Clues:

1. No child has a pet that starts with the same letter as their name.

2. Dave does not have a pet that lives in the water.

3. Molly is allergy to smoke.

4. Wendy loves to fly.

Troll

Water horse

Mermaid

Dragon

Dave

Wendy

Molly

Tracy

Conclusion:

2. Five pirates buried their secret treasures on an island. Each treasure is buried near 

a different landmark. Where did each pirate bury his or her treasure?

Clues:

1. There are no tall trees near Bart or Bonnie's treasure.

2. Jean and Bonnie did not bury their treasure near the rock.

3. Sam buried his treasure near water.

4. Bart did not bury his treasure near the hill.

5. The water near Roger’s treasure is calm and quiet.

Giant rock

Tall tree

Jolly pond

Wild 

waterfalls

Green hill

Roger

Bonnie

Bart

Sam

Jean

Conclusion:

A. List the members of the following sets

1. {x| x is real numbers and x2 = 1}

2. {x| x is an integer and -4 < x ≤ 3}

B. Use set builder notation to give description of each of these sets.

1. {a, e,i ,o, u}

2. {=2, -1, 0, 1, 2}

C. Let A= (a, b, c), B = (x, y) and C = (0, 1)

Find:

1. A U C

2. C x B

3. B – A

4. (A ∩ C) U B

D. Find these terms of the sequence (An}, where An = 2(3)n + 5

1. A0

2. A5

3. A3

4. 8th term

5. 2nd term

6. Sum of the sequence

E. Given the following set:

2. X = {-1, 0, 1, 2, 3, 4, 5} defined by the rule (x, y) ∈R if x ≤ y

F. List the elements of R

G. Find the domain of R

H. Find the range of R

I. Draw the digraph

J. Properties of the Relation

Let A = {1, 2, 3, 4, 5, 6, 7} and

R = {(x, y) | x –y is divisible by 3}

Show that R is an equivalence relation. Draw the graph of R.

In how many ways can six boys and four girls be arranged in a straight line so that no two girls are ever together.

1. Given the following:

• g: "You can graduate."
• m: "You owe money to the college."
• r: "You have completed the requirements of your major."
• b: "You have an overdue book."

Translate "You can graduate only if you have completed the requirements of your major, you do not owe money to the college, and you do not have an overdue book." into a propositional logic.

2. Show that  are logically equivalent. (15 points)

3. Show, by the use of the truth table (truth matrix), that the  is a contradiction. (15 points)

• <i class="add" style="width: 20px; height: 20px; display: inline-block; vertical-align: top; position: relative; overflow: hidden; margin: 0px 6px 0px 0px;"></i>Prepare answer

^{A. Find the indicated sets.}

2.ℙ({1,2,3,4})

6. ℙ({1,2})x ℙ({3})

10.  {Xϵ ℙ({1,2,3}):│X│≤1}

B. Suppose that │A│=m and │B│=n. Find the following cardinalities.

14. │ ℙ(ℙ(A))│

18. │ ℙ(Ax ℙ(B))│

A.     List all the subsets of the following sets.

2. {1,2,Ø}

8. {{0,1},{0,1,{2}},{0}}

B. Write out the following sets by listing their elements between braces.

12. {X:X ⊆{3,2,a} and │X│=1

C. Decide if the following statement is tru or false. Explain.

14. R²⊆R³

A.     Write out the indicated sets by listing their elements between braces.

2. Suppose A={π,e,0} and B={0,1}

(a)   AxB

(b)  BxA

(c)   AxA

(d)  BxB

(e)  ØxB

(f)    (AxB)xB

(g)   Ax(BxB)

(h)  AxBxB

4. {nϵZ:2<n<5}x{nϵZ:│n│=5}

B. Sketch these Cartesian products on the x-y plane R².

10. {-1,0,1}x{1,2,3}

12. [-1,1]x[1,2]

14. [1,2]x{1,1.5,2}

16. [0,1]x{1}

A. Write each of the following sets by listing their elements between braces.

2. {3x+2:xϵZ}

4. {xϵN:-2<x≤7}

6. {xϵR:x²=9}

8. {xϵR:x³+5x²=-6x}

10. {xϵR:sin πx=0}

B. Write each of the following sets in set-builder notation.

18. {0,4,16,36,64,100,…}

20. {…,-8,-3,2,7,12,17,…}

22. {3,6,11,18,27,38,…}

24. {-4,-3,-2,-1,0,1,2}

26. {…,1/27,1/9,1/3,1,3,9,27,…}

C. Find the following cardinalities.

30. │{{1,4},a,b,{{3,4}},{Ø}}│

32. │{{{1,4},a,b,{{3,4}},{Ø}}}│

34. │{xϵN:│x│<10}│

36. │{xϵN:x²<10}│

38. │{xϵN:5x≤20}│

D. Sketch the following sets of points in the x-y plane.

40. {(x,y):xϵ[0,1],yϵ[1,2]}

42. {(x,y):x=2,yϵ[0,1]}

44. {(x,x²):xϵR}

46. {(x,y):x,yϵR,x²+y²≤1}

48. {(x,y):x,yϵR,x>1}

50. {(x,x²/y):xϵR,yϵN}