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Set theory and logic problem set

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  • 1. SET THEORY AND LOGIC
    PROBLEM SET NUMBER 1
    RICHARD B. PAULINO
    1
  • 2. RICHARD B. PAULINO
    2
    Problem # 1.
    Use the set notation for the following statements:
    a. The set of all pairs of numbers a and b, being a an integer and b a real number
    A = { (a,b)|a Є b Є R }
    b. x is a member of the set A
    B = { x|x Є A }
    c. A is the set of the vowels of English Alphabet
    A = { x|x is a vowel in the english alphabet }
  • 3. RICHARD B. PAULINO
    3
    Problem #2.Write the following in tabular form
    a. A = { x|x+3=9}
    A= { 6 }
    b. D= { x|x is a number of two digits divisible by 7}
    D = A= { 14, 21, 28, 35, 42, 49, 56, 63, 70, 77, 84, 91, 98 }
    Problem #3.Which of the following sets are equal?
    a. { x|x is a letter in the word mathematics}
    b. A set containing m,a,t,h,e,i,cand s
    c. A set of letters containing the word manuscript.
    a and b are equal sets
    Problem #4. Which of these sets is/are null set(s)?
    A = { x|x is a a letter after Z in the alphabet}
    B = { x|x2 = 4 and x = 1}
    C = { x| x + 9 = 9}
    D = { x|x≥4 and x<1 }
    Sets A, B and D are null sets
  • 4. RICHARD B. PAULINO
    4
    Problem #5. Which sets are subsets of the others?
    A = { x|x is a Quadrilateral}
    B = { x|x is a rhombus}
    C = { x|x is a rectangle}
    D = { x|x is a square}
     
    Properties of Quadrilaterals and Parallelograms
    - All parallelograms are Quadrilaterals
    Square, Rectangle and Rhombus are parallelograms
    B ⊆ A ᴧ B A
    C ⊆ A ᴧ C A
    D ⊆ A ᴧ D A
    - All squares are rectangles/rhombus, but not all rectangles/rhombus are squares.
    C ⊆ D C D ᴧ D ⊆ C but D C
    B ⊆ D B D ᴧ D ⊆ B but D B
  • 5. RICHARD B. PAULINO
    5
    Problem #6. State whether each of the following sets is infinite or finite.
    a. The set of lines parallel to y-axis
    b. The set of articles through the origin
    c. The roots of the equation (x-1)(x-2)(x-3)
    d. The set of number & which are multiple of 7
    a. infinite
    b. finite
    c. infinite
    d. infinite
     
     
    Problem #7. Write true or false for the following statements if A = {1, {4,5}, 4}
    a. {4,5} A a. FALSE
    b. {{4,5}} A b. TRUE
  • 6. RICHARD B. PAULINO
    6
    Problem #8. If A = {-1 ≤ X ≤ 1}
    B = {1 < X < 0}
    C = {0 ≤ X < 2}
    Find:
    A U C b. A ∩ B c . A U B U C d. (A U B) ∩ C e. B∩C
    Solution:
    A = {-1 ≤ X ≤ 1}
     B = {1 < X < 0}
     C= {0 ≤ X < 2}
     
    a. A U C = { -1 ≤x <2}
    b. A ∩B ={ -1 ≤x <0}
    c. A U B U C = { -1 ≤x <2}
    d.(A U B) ∩C = { 0 ≤x ≤ 1}
    e. B∩C = { }
     
  • 7. RICHARD B. PAULINO
    7
     
    Problem #9. Out of 40 students 30 can jump, 27 can play football and 5 can do neither. How many students can jump and play football?
     
    Solution:
     
    Let:
    U = 40 students
    J U G = 30 students that can jump
    F U G = 27 students can play football
    N = 5 students that can do neither
    G = number of students that can do both (jump and play football).
     
  • 8. RICHARD B. PAULINO
    8
    Problem #9 cont’n.
    J U G U F= U – N 40 – 5 = 35
    J + G = 30
    F + G = 27
    J + G + F = 35
    J + G + F = 35
    J + G = 30
    F = 5 number of students that can play football only
     
    J + G + F = 35
    F+ G = 27
      J = 8 number of students that can jump only
     
    Hence, J + G + F = 35
    8 + G + 5 = 35
    G = 35 – ( 8 + 5 )
    G = 22 number of students that can both jump and play football
  • 9. RICHARD B. PAULINO
    9
    Problem #10.
    In a certain school 80 students went to the zoo with hamburger, milk and cake with breakdown us follows:
    2 had hamburger and milk
    6 had cake & milk
    12 had cake & hamburger
    36 had cake
    22 had milk
    a. How many had nothing? b. How many had cake?
    c. How many had hamburger and cake? d. How many had hamburger and milk
    e. How many had hamburger only? f. How many had milk only?
    g. How many had milk and cake?
  • 10. RICHARD B. PAULINO
    10
    Problem #10. Cont’n.
    Solution:
     
     
     
     
     
     
     
     
    a. How many had nothing? b. How many had cake only?
    IUI - I A U B U C I + I A П B П C I ICI - I A П C I - I B П C I
    = 80 – ( 14 + 22 + 36 ) + (20) = 36 – 12 – 6
    = 28= 18
    c. How many had hamburger and cake? d. How many had hamburger and milk?
    = 12 = 2
    e. How many had hamburger only? f. How many had milk only?
    = 0 IBI - I B П C I - I B П C I
    g.How many had milk and cake? = 22 – 6 – 2
    = 6 = 14
  • 11. 11
    THANK YOU
    VERY MUCH!
    from
    Richard Josie
    XyraXyron