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Set Theory

Set Theory. Using Mathematics to Classify Objects. 2.2 Comparing Sets Objectives:. Determine when sets are equal Know the difference between the relations of subset and proper subset Use Venn diagrams to illustrate set relationships Determine the number of subsets of a given set

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Set Theory

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  1. Set Theory Using Mathematics to Classify Objects

  2. 2.2 Comparing Sets Objectives: • Determine when sets are equal • Know the difference between the relations of subset and proper subset • Use Venn diagrams to illustrate set relationships • Determine the number of subsets of a given set • Distinguish between the ideas of “equal” and “equivalent” sets

  3. Set Equality • Examples • A = { 2, 4, 6 } , B = { 2, 5, 6 } • n(A) = _____ ? n(B) = _____ ? • A ___ B • A = { 2, 3, 4, 5 } , B = { x : x is a natural number between 1 and 6 } • n(A) = ______ ? n(B) = ______ ? • A ____ B

  4. Subsets • If, • then A “is contained in” B. • If , • then __________________________________________ • Example: • A = { 2, 4, 6 } , B = { 2, 3, 5, 6 } • Is A a subset of B? A ____ B • Why? __________________________________ • Question: • Is { } a subset of B? ____

  5. Proper Subsets If A is a proper subset of B , then n(A) < n(B) So: every element of A is in B why?____ not every element of B is in A. why?____ Example Let A = { 2, 6 } How many subsets does A have ? ___ How many proper subsets does A have ? ___

  6. Venn Diagrams A Venn diagram is used to visualize relationships among sets. • The rectangle represents the Universal set (We are thinking “inside the box”.) • Regions inside the rectangle represent various subsets of the Universal set. • What is the relationship between sets A and B in the Venn diagram above? • ____________________________

  7. How many Subsets? • Let A = { 2, 6 } n(A) = _____ List the subsets of A: • Let B = { 2, 6, x } n(B) = _____ Is every subset of A also a subset of B? ____ List the other subsets of B: • Let C = { 2, 6, x, y } n(C) = _____ Is every subset of B also a subset of C? ____ How many other subsets are there? ____

  8. How many Subsets, cont. • Consider a general set X with the given number of elements. How many subsets does it have? n(X) = 0 ________ n(X) = 1 ________ n(X) = 2 ________ n(X) = 3 ________ • Each time that the number of elements is increased by 1, the number of subsets _______________ .

  9. How many Proper Subsets? • How many subsets exist for the given set? • How many proper subsets exist for set A? _____ • A set that has k elements has ____________ proper • subsets.

  10. Equivalent Sets • The sets {1, 2, 3} and {A, B, C} are equivalent because they both have 3 members. • * Show a 1-to-1 correspondence between the sets. • Are the set of natural numbers and the set of integers equivalent? ____ • “Count” the integers! • * Show a 1-to-1 correspondence between the sets.

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