Ang aking kontrata: Ako, si, ay nangangakong magsisipag mag-aral hindi lang para sa aking sarili kundi para rin sa aking pamilya, para sa aking

Size: px

Start display at page:

Download "Ang aking kontrata: Ako, si, ay nangangakong magsisipag mag-aral hindi lang para sa aking sarili kundi para rin sa aking pamilya, para sa aking"

Cathleen Copeland
5 years ago
Views:

1 Ang aking kontrata: Ako, si, ay nangangakong magsisipag mag-aral hindi lang para sa aking sarili kundi para rin sa aking pamilya, para sa aking bayang Pilipinas at para sa ikauunlad ng mundo.

2 Mathematics Division, IMSP, UPLB

3 Learning Objectives: Set Relations Upon completion you should be able to identify set relations such as equality subset and superset equivalence Mathematics Division, IMSP, UPLB

4 Equal Sets Set Relations Two sets A and B are equal if and only if they have the same elements. Example: Let A = {1, 3, 5, 7, 9} and B = {3, 5, 1, 9, 7}. Are all the elements in A also in B? Are all the elements in B also in A? YES! Therefore A and B have the same elements. Sets A and B are. Mathematics Division, IMSP, UPLB

5 Equal Sets Set Relations If sets A and B are equal, we write A=B. Otherwise, we write A B. Example: Let C = {1, 2, 3, 4} and D = {1, 2, 2, 3, 4, 4}. Are all the elements in C also in D? Are all the elements in D also in C? YES! Therefore C = D. Mathematics Division, IMSP, UPLB

6 Equal Sets Set Relations REMEMBER: 1. IN A SET, IT IS CUSTOMARY TO LIST AN ELEMENT ONLY ONCE. 2. IN A SET, THE ORDER OF LISTING THE ELEMENTS DOES NOT MATTER. 3. TWO SETS ARE EQUAL IF AND ONLY IF THEY HAVE THE SAME ELEMENTS. Mathematics Division, IMSP, UPLB

7 Subsets and Supersets Set Relations If all elements of set A are also elements of set B, we say, A is a subset of B (or B is a superset of A). Example: Let S be the set of all students in this room. Let B be the set of all boys in this room. Let G be the set of all girls in this room with age less than 16. Is B a subset of S? Is G a subset of S? Mathematics Division, IMSP, UPLB

8 Subsets and Supersets U Set Relations B G S Mathematics Division, IMSP, UPLB

9 Subsets and Supersets TIME TO THINK! 1. Is U always a superset? Set Relations 2. Is a set a subset of itself? 3. Is a set a superset of itself? 4. Do you think { } is a subset of any set? 5. Do you think { } is a superset of any set? Mathematics Division, IMSP, UPLB

10 Subset Set Relations If A is a subset of B, we write A B. In our previous example, S is the set of all students in this room. B is the set of of all boys in this room. G is the set of all girls in this room with age less than 16. B S and G S. However, B is not a subset of G. Why? In this case, we write B G. Mathematics Division, IMSP, UPLB

11 Subset Set Relations Suppose A is a non-empty set. If A B and A B, then we call A a proper subset of B. If A = {, } and B = {,,, } then A is a proper subset of B and we may write A B or A B. Mathematics Division, IMSP, UPLB

12 Subset Set Relations There are only two improper subsets. The empty set and the set itself. If A = {, } and B = {,,, } then {} and A are improper subsets of A and we write {} A and A A. {} and B are improper subsets of B and we write {} B and B B. Mathematics Division, IMSP, UPLB

13 Subset Set Relations SUBSETS OF SET J PROPER IMPROPER Empty set Set J Mathematics Division, IMSP, UPLB

14 Subset Set Relations Always True, Sometimes True or False: Let A, B, and C be sets. 1. A A 2. A A (Reflexive Property) 3. If A B then B A 4. If A B then B A 5. If A B and B C then A C (Transitive Property) 6. {} A 7. {} {} 8. A U Mathematics Division, IMSP, UPLB

15 Subset Set Relations Determine if proper or improper subset of {1,2,3,4,5}: 1. {1,2} 2. {4} 3. {1,2,3,4,5} 4. {2,3,4} 5. {} Mathematics Division, IMSP, UPLB

16 Set Relations ALTERNATIVE DEFINITION OF EQUALITY OF SETS A=B if and only if A B and B A. Mathematics Division, IMSP, UPLB

17 Set Equivalence Set Relations What can you observe about the following pairs of sets? A = {1, 2, 3, 4, 5} B = {a, e, i, o,u} C = {guava, melon, avocado} D = {do, re, mi} Mathematics Division, IMSP, UPLB

18 Set Equivalence Two sets are in 1-1 correspondence if it is possible to pair each element of A with exactly one element of B, and each element of B with exactly one element of A. It follows that A and B have the same size or number of elements. When two sets A and B are in 1-1 correspondence, we say they are equivalent and we write A B. Thus, in our example, A B and C D. Is A C? Why? Mathematics Division, IMSP, UPLB Set Relations

19 Set Relations Set Equivalence 1-1 Correspondence 1 a 20 b 3 c THEY ARE EQUIVALENT Mathematics Division, IMSP, UPLB

20 Set Relations Set Equivalence 1-1 Correspondence 1 a 20 b 3 c THEY ARE EQUIVALENT Mathematics Division, IMSP, UPLB

21 Set Equivalence Set Relations Does the following exhibits 1-1 Correspondence? Are they equivalent? a b c d Mathematics Division, IMSP, UPLB

22 Set Equivalence Set Relations Does the following exhibits 1-1 Correspondence? Are they equivalent? a b c d Mathematics Division, IMSP, UPLB

23 Example Is there a one-to-one correspondence between the set of days in a week and the set of counting numbers from 2 to 8? YES M T W Th F Sa Su THEY ARE EQUIVALENT

24 Example Is there a one-to-one correspondence between the set of days in a week and the set of months in a year? NO THEY ARE NOT EQUIVALENT

25 Example Is there a one-to-one correspondence between the set of even counting numbers and the set of odd counting numbers? YES THEY ARE EQUIVALENT

26 Set Equivalence Set Relations Time to think: 1. Are all equal sets equivalent? 2. Are all equivalent sets equal? 3. Can a set be equivalent to any of its subsets? 4. Can a set be equal to any of its subsets? Mathematics Division, IMSP, UPLB

27 Set Relations Exercise For each of the sets listed below, tell which are equivalent and which are also equal. 1. The set of distinct letters in the word katakataka 2. The set {a,k,t,k} 3. The set of distinct letters in the word tatak 4. The set {k,t,a} 5. The set {k,a,r} Mathematics Division, IMSP, UPLB

28 Summary Set Relations In this section, we learned When two sets are equal or not When a set is a subset or superset of another When two sets are equivalent or not Mathematics Division, IMSP, UPLB

30 QUESTION: IN SET THEORY, Is countable and finite the same? Is uncountable and infinite the same?

31 CARDINALITY Cardinality of a set is a measure of the size or the number of elements of the set. What is the cardinality of { }?

32 COUNTING, 1-1 CORRESPONDENCE AND CARDINALITY SET OF NATURAL/ COUNTING NUMBERS SET J a b c

33 COUNTING, 1-1 CORRESPONDENCE AND CARDINALITY The cardinality of set J is J = n(j) = 3

34 CARDINALITY AND SET EQUIVALENCE Two sets are equivalent if they have the same cardinality.

35 COUNTABLE SET A set where you can have 1-1 correspondence with a subset of natural/counting numbers is countable. Otherwise, it is uncountable.

36 COUNTABLE SET Question: Is the set of positive even integers countable?

37 COUNTABLE SET YES! This is called countably infinite!

38 FINITE AND INFINITE SET A set is finite if it has a cardinality equal to a counting/natural number. Example: n(j)=3 All finite sets are countable!

39 FINITE AND INFINITE SET A set is finite if your counting ends. A set is finite if after listing all the elements, there is a last element. Example:{a,b,c,d,e} Counterexample:{1,2,3,4,5, }

40 FINITE AND INFINITE SET A set is infinite if it is not finite. Example: The set of natural/counting numbers {1,2,3, } has infinitely many elements, hence it is an infinite set. But, it is countable!

41 FINITE AND INFINITE SET Example: The set of positive even integers does not have cardinality equal to a natural/counting number, so it is infinite. But it can have a 1-1 correspondence with the set of natural/counting number so it is countable.

42 FINITE AND INFINITE SET Example of uncountable infinite set: The set of real numbers (because we cannot have a 1-1 correspondence between the set of reals and the set of counting numbers)

43 Summary Infinite sets can be countable or uncountable. All uncountable sets are infinite sets. But not all infinite sets are uncountable.

44 FINITE AND INFINITE SET Exercise: Determine if FINITE or INFINITE, and if COUNTABLE or UNCOUNTABLE 1) Set of points in a circle 2) Set of counting numbers between 1 and ) Set of real numbers between 0 and 1 4) The set of all sands in Boracay beach

45 FYI The cardinality of the set of natural/counting numbers is ℵ 0 (aleph-null). The cardinality of the set of real numbers is ℵ 1 (alephone) or c (for continuum). Note: ℵ 0 and ℵ 1 are not real numbers.

46 TRIVIA Can a set be equivalent to one of its proper subset? YES! When would this happen? If the set is infinite. Can you give an example? N~E.

47 The concept of INFINITY is mysterious. You may read some articles about this concept on the internet

MATH 17: College Algebra and Trigonometry. Lecturer: Jomar Fajardo Rabajante 1st Semester AY Math Division, IMSP, UPLB

MATH 17: College Algebra and Trigonometry. Lecturer: Jomar Fajardo Rabajante 1st Semester AY Math Division, IMSP, UPLB Ang aking kontrata: Ako, si, ay nangangakong magsisipag mag-aral hindi lang para sa aking sarili kundi para rin sa aking pamilya, para sa aking bayang Pilipinas at para sa ikauunlad ng mundo. MATH 17: