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Sets: An introduction

Sets: An introduction. Coming up… Probability. In our study of probability, two things will be of great use to us: our understanding of set operations our counting ability. Definition. A set is a well-defined collection of objects, called elements. Remark:

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Sets: An introduction

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  1. Sets: An introduction

  2. Coming up… Probability • In our study of probability, two things will be of great use to us: • our understanding of set operations • our counting ability

  3. Definition A set is a well-defined collection of objects, called elements. Remark: Not every collection of objects is a set. There must be a definite way of telling what is and what is not in the set.

  4. Example #1 Determine which of the following collections are sets. 1.The collection of states in the US. 2.The collection of great US presidents.

  5. Set Representation We consider 3 ways of defining a set. • A set may be defined by listingits elements. For instance, the set A of letters in the word “beloved” may be written as A = {e, b, l, o, d, v} Sets are often named with capital letters. Order of elements doesn’t matter; no duplicates.

  6. 2. A set may be defined by describinga common feature of all of itselements. For instance, consider the set B given by B = {x | x is a positive integer less than 5} This is read: “x such that x is a positive integer less than 5” Clearly, the elements of B are 1, 2, 3, and 4.

  7. 3. A set may be defined by drawinga Venn diagram. For instance, the set C may be represented by the circle shown below inside a rectangle. C

  8. Example #2 • Find the set of all the possible outcomes: • when you flip a penny. • when you toss a die.

  9. a) Let A be the set of the possible outcomes when you flip a penny. Then A = {h, t}. b) Let B be the set of the possible outcomes when you toss a die. Then B = {1,2,3,4,5,6}.

  10. Set Operations Intersection The intersection of two sets A and B is the set of elements that are in both sets. It is denoted by A ∩ B.

  11. Union The union of two sets A and B is the set of all elements of A or B. It is denoted by A U B. B A

  12. If every element of a set A is also an element of a set U, then say that A is a subset of U. For instance, the set A = {b, e, t} is a subset of the set U = {b, e, a, s, t}.

  13. Complement of a Set Let A be a subset of U. Then we define the complement of A in U as the set of all elements of U not in A. It is denoted by A'. A U

  14. Remarks: • The set with no element is called the empty setand is denoted by • The number of elements in a set A is denoted by n (A). • A set may be • finite, like the set of letters of the English alphabet or • infinite like the set of positive whole numbers.

  15. Example #3 Given the sets: U = {h, i, m, e, a, t, l, o, v, r, s} A = {m, a, t, h} B = {h, a, t, e} C = {l, o, v, e} Find: 1) B ∩ C 2) A U B 3) n(U) 4) B’ 5) A ∩ C 6) n(A ∩ C)

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