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Instructor:
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Mr. Rameel Rizvi
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Attendance:
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Aiden, Omar, Ayyan
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Material Covered:
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Special topic: introduction to basic set theory
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Problems Solved in Class:
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Examples of set unions, intersections, differences, and symmetric differences. Also briefly talked about Euler's number, e, which is approximately 2.71828, and we saw that the function f(x) = e^x has some very interesting properties.
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Concepts:
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- a set is a collection of objects; there is no restriction whatsoever on these objects
- elements of a set do not repeat
- we use brackets to define a set, separating the listed elements with commas (for example, A = {1,2,3} is a set containing exactly the numbers 1, 2, and 3)
- a null set, or empty set (i.e. a set containing no elements), is denoted by { } or ∅
- the union of two sets A and B, denoted A ∪ B, is a set containing all elements from A and all elements from B
- the intersection of two sets A and B, denoted A ∩ B, is a set containing those elements that belong to both A and B, and no other elements
- the difference of two sets A and B, denoted A \ B, is a set containing those elements that belong to A but not to B, and no other elements
- the symmetric difference of two sets A and B, often denoted A ⊕ B, is a set containing those elements that belong to either A or B, but not to A ∩ B, and no other elements
- the cardinality of a set A, denoted |A|, is the number of elements that are members of A
- a proper subset or strict subset S of a set A, denoted S ⊂ A, is a set that is strictly smaller (i.e. has a smaller cardinality) than A and all of whose elements are also members of A
- we may allow a subset S to have the same cardinality as A, and denote this possibility by S ⊆ A (so S could be the same size as or smaller than A, and we simply call it a subset rather than a strict subset or proper subset)
- the power set of a set A, denoted P(A), is the set of all subsets of A
- if |A| = k, then |P(A)| = 2^k (we saw this in class by considering binary strings of length k and observing that we have 2 choices for each digit of the string, either 0 or 1)
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Student Difficulties:
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Students were slightly familiar with sets and operations union and intersection, but the rest was new and was understandably not the easiest to grasp, but I was quite happy with what was learned.
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Homework:
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Let U be the set of all lower-case letters of the English alphabet. Let
A = {a,b,f,h,w,y} and B = {w,c,h,q,e,m}. Answer all of the following:
1) What is |U|?
2) Write the smallest subset of U.
3) How many proper subsets of U are there?
4) How many proper subsets of U are there that have maximal cardinality?
5) What is U \ {a,e,i,o,u}? What is this set commonly called?
6) Write A ∪ B.
7) Write A ∩ B.
8) Write A \ B.
9) Write B \ A.
10) Write A ⊕ B.
11) Write P(U ∩ {x,y,z}).
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Notes:
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