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

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