Recap/HW for 02/12/17 Class

Instructor:	Mr. Rameel Rizvi
Attendance:	Aiden, Omar, Ayyan
Material Covered:	Special topic: introduction to basic set theory
Problems Solved in Class:	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.
Concepts:	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)
Student Difficulties:	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.
Homework:	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}).
Notes:

Instructor:	Mr. Rameel Rizvi
Attendance:	Aiden, Omar
Material Covered:	Pre-Algebra Chapter 11, 12 (up til and including 12.2)
Problems Solved in Class:	Chosen exercises from chapters 11 and 12
Concepts:	* Covered perimeter of bounded figures.  For a rectangle, the total perimeter length is given by 2(l + w) where l is the length, w the width For a circle, this is the circumference, whose length is 2πr with r the radius * Covered area of bounded figures For a rectangle, the total area is given by l*w square units, where l is the length, w the width For a circle, this is πr^2 with r the radius For a triangle, this is (1/2)bh where b is the base length and h is the height * Triangle inequality: For any 3 points A,B,C on a plane, AB + BC >=  AC, and this is only equal for a line * Pythagorean Theorem: For a right triangle with sides a,b,c we have a^2 + b^2 = c^2    - Proved this using triangles enclosed in a square * Special triangles    - 45-45-90 (side lengths a and a*sqrt(2))    - 30-60-90 (side lengths a, a*sqrt(3), 2a) * Briefly covered a quadrilateral known as a rhombus, whose defining property is all sides having equal length. The diagonals of a rhombus bisect the vertex angles (split them in half), and they intersect  perpendicularly (forming right angles). The area of a rhombus can be found by multiplying the lengths of the diagonals and halving the result.
Student Difficulties:	Students seemed familiar to some extent with the shapes covered, but there may have been trouble understanding the proofs that were presented (such as for Pythagoras' Theorem or for the area of a rhombus).
Homework:	Chapter 11: Read chapter summary. Problems: 11.40, 11.45, 11.47, 11.50 Chapter 12 Problems: 12.1.2, 12.1.5, 12.2.6
Notes: