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# MELJUN CORTES Discrete MATH Part I SETS

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MELJUN CORTES Discrete MATH Part I SETS

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### MELJUN CORTES Discrete MATH Part I SETS

1. 1. DISTRUC – Discrete Structures (Discrete Mathematics)MELJUN CORTES
2. 2. SETSSET is a well-defined collection of distinct objects. Awell-defined set means that is possible to determinewhether an object belongs to a given set. The objectsare called MEMBERS or ELEMENTS.Ex. 1. Collection of vowels in the english alphabet 2. Collection of even numbers 3. Collection of brand mobile phones 4. Collection of favorite color 5. Collection of good singers
3. 3. SETS2 WAYS OF DESCRIBING A SET:1. TABULAR OR ROSTER FORM – a method of describing a set where elements are separated by commas and enclosed by braces.2. RULE FORM – method of describing a set which makes use of the description.. { x | … }
4. 4. SETSIllustration: RULE ROSTER1. {x/5|x is a whole {0, 1/5, 2/5, 3/5} number less than 4}2. {x|x is an even integer {2, 4, 6} between 0 and 8}3. {1/x|x є N} {1, 1/2, 1/3, …}
5. 5. SETSNote: A set which contains no element is called anempty set { } or null set Ǿ.KINDS OF SETS1. Equal Sets - if both sets has the same elementsEx. A = {1, 2, 3} and B = {2, 1, 3}2. Equivalent Sets - if both sets have the samenumber of elementsEx. A = {a, b, c} and B = {1, 2, 3}
6. 6. SETS3. Finite Sets - if it contains only a countable numberof elements.Ex. A = {a, b, c, d}4. Infinite Sets - if the counting of elements has noend.Ex. N = {1, 2, 3…}5. Universal Sets - the totality of elements underconsideration.Ex. A = {1, 2, 3}, B = {3, 4, 5} U = {1, 2, 3, 4, 5}
7. 7. SETS6. Joint Sets - sets that have commonelements.Ex. A = {4, 5, 6} and B = {6, 10, 11}7. Disjoint Sets - if two sets have no commonelementsEx. A = {3, 4, 5} and B = {h, I, j}
8. 8. SETSOperation on Sets The operation on sets behave in amanner somewhat similar to the basicoperations on numbers. The Venn Diagram ofsets makes use of a rectangle representing the“Universal Set” and circles are subjects whichmay be shaded under consideration.
9. 9. SETS4 BASIC OPERATIONS ON SETS1. Union (A U B) – set of all elements found in A or in B or both.2. Intersection (A ∩ B ) – set of all elements common to A and B.3. Complement (A’) – set of all elements in the universal set but not found in A.4. Difference (A – B / B – A) – sets of all elements found in A but not in B or vice versa.
10. 10. SETSLaws of Sets:1. Commutative Law – the order in w/c the sets are taken does not affect the result.2. Associative Law – the grouping in w/c the sets are taken does not affect the result.3. Identity Law – a set separated to another called the identity gives the set itself.4. Inverse or Complement Law – these laws involves inside and outside of a set.5. Distributive Law – this law involves three sets with two different operation, distributing the first operation over the second one.6. De Morgan’s Law – the complement of union sets A and B is the intersection of the respective complement of A and B. The statement holds upon interchanging the words union and intersection.
11. 11. SETSExamples:Given: U = {1, 2, 3, … 15} A = {odd numbers between 0 to 10} B = {3x + 1 | x € Z}Required:1. AUB =2. A∩B =3. A’ =4. B–A =5. (A U B)’ =
12. 12. SETSApplication:1. In a survey involving 150 different factories, it was found out that: 70 purchased brand A 75 purchased brand B 95 purchased brand C 30 purchased brand A and B 45 purchased brand A and C 40 purchased brand B and C 10 purchased brand A, B, and C
13. 13. SETSQuestions:1. How many had purchased A only? ____2. How many had purchased at least two brands? ____3. How many hadn’t purchased any of the products? ___
14. 14. SETS2. In an excursion at Pagsanjan Falls, 180 studentsbrought sandwiches, drinks, and cans as follows: 50 students brought sandwiches 30 students brought drinks 30 students brought cans 18 students brought cans and drinks 15 students brought sandwiches and cans 8 students brought sandwiches and drinks 5 students brought sandwiches, drinks, and cans
15. 15. SETSQuestions:1. How many had nothing? ____2. How many had cans only? ____3. How many had sandwiches and drinks? ___4. How many had sandwiches and cans? ___5. How many had drinks and cans but not sandwiches? ___6. How many had at least 2 items? ___7. How many had at most 2 items? ___
16. 16. SETS End of Part I – Sets