Set theory for teachers

Set theory for teachers

Set theory for teachers

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Presentation Transcript

1. Set theory for teachers MA118 Summer 2008 McAllister

2. Background of set theory • Georg Cantor (1845-1918). • Big component of ‘new math’ curriculums that were pervasive in 60’s and early 70’s. • The most fundamental of concepts upon which all of mathematics is grounded • Important for teachers to understand the language and basics of sets. • Understanding teachers’ manuals and research • Add dept to your understanding of math

3. Vocabulary • set- a collection of objects • elements, or members - things in a set • subsets- a part of another set • proper subsets- a subset that is not equal to the set it is a part of • Universal set – The set of all possible elements under consideration

4. More terms • empty set(null)- set with no members • well-defined sets-sets that are described clearly so there's no question of membership • Finite sets – sets that can be counted • Infinite set – sets that cannot be counted

5. Examples of number sets • natural numbers- {1, 2, 3, 4, …} (counting numbers) • whole numbers- {0, 1, 2, 3, 4, …} • integers – The set of whole numbers and their opposites. • rational numbers- numbers that can be written as a fraction.

6. More examples of number sets • irrational numbers- numbers that can not be written as a fraction, OR non-repeating decimals, non-terminating decimals • real numbers- numbers that can describe a distance, numbers that form a one-to-one correspondence with the points on a number line. Composed of the rationals and irrationals. • evens-divisible by 2 • odds-not evenly divisible by 2 • prime- a number with exactly 2 factors, 1 and itself • Is one a prime? No. • composite- a number with more than 2 unique factors. • multiples- exp: multiples of 2={2, 4, 6…}

7. Notation – very important • Words • The set of all teachers in the room. • The set of even numbers between 1 and 9 • List Elements • A = {McAllister} • B = {2, 4, 6, 8} • Set Builder Notation • A = { x1 x is a certified teacher} • B = {x1x is even and 1 < x < 9 }

8. Other notation and definitions • It’s time to move to the document camera for this part.

9. Venn diagram problems Twenty-four dogs are in a kennel.  Twelve of the dogs are black, six of the dogs have short tails, and fifteen of the dogs have long hair.  There is only one dog that is black with a short tail and long hair.  Two of the dogs are black with short tails and do not have long hair.  Two of the dogs have short tails and long hair but are not black.  If all of the dogs in the kennel have at least one of the mentioned characteristics, how many dogs are black with long hair but do not have short tails?

10. Disjoint set problem • In a group of 37 people, 18 are neither women nor lawyers. Ten are women and 13 are lawyers. How many lawyers in the group are not women?

11. One more In a survey of 6500 people, 5100 had a car, 2280 had a pet, 5420 had a T.V., 4800 had a T.V. and a car, 1500 had a T.V. and a pet, 1250 had a car and a pet, and 1100 had a T.V., car, and pet. How many people had no car, no T.V. and no pet?