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Modern Algebra For Grade 5 What are some words that come to mind when you hear the word ALGEBRA? From the grade 5 class: • Math • Numbers • Operations • Algae • Measuring Today, we are going to learn about sets. Does anyone know what a set is? From the grade 5s: • A question • Math set • Set of apples • Set of shoes What is a set? • Mathematicians have agreed not to define a set. They just hope that when people hear “the set of all Canadians” or “the set of all grade five students”, everyone know what that means and who/what it includes. • A set is a subset of a set if everything in the subset is in the set. • Ex) Set is students at Chedabucto Place, and a subset is grade 5 students Things to assume about sets: • A set S is made up of elements, and if a is one of these elements, we shall denote this by a S. • There is exactly one set with no elements. It is the empty set and is denoted by . • We may describe a set either by giving a property that characterizes the elements, or by listing all the elements. • A set is well defined. If S is a set and a is some object, then either a is in S, or a is not in S Subsets • A set B is a subset of a set A if every element of B is in A. • For a set A, A and the empty set are subsets of A. • A is the improper subset of A. • All other subsets are proper subsets. • A subset is denoted by: B ≤ A or A ≥ B Activity • Our set today is the whole grade 5 class. • Lets arrange into subsets based on hair colour! • So, the whole grade 5 class is the set. Smaller subsets can be formed from parts of the whole set. Next, lets form into subsets based on your favourite colour out of: • • • • Red Blue Pink Purple • • • • Green Yellow Black Orange Empty Sets • A subset is still considered a subset even if there is nothing in it. • This is called the empty set • From our activity, if nobody picked orange as their favourite colour, orange would be an empty subset. Cycles • A cycle is a way to describe the order of elements in an orbit. • Equivalence classes are called orbits. An orbit is just a subset of the original set. Activity • We are going to learn how cycles work by doing a hands on activity. • Everyone will stand in a circle. • I will divide the group into cycles. • Each cycle will have an order they must follow. • Each cycle will have it’s own ball which is to be passed among the cycle. • Students pass the ball only to the person who comes after them. Cycles • Notice how each ball is passed within the same people the whole time. • The ball never goes to anybody else in the circle. • For example, someone who started with the blue ball was never thrown the yellow ball. Some Things Students Learned: “If you separate all the kids in grade five into groups with the same hair colour, each set is a subset. Ex) brown is a subset, blonde is a subset, black is a subset” “If the set was the NHL team Pittsburgh, the subset might be all the forwards.” “A set can be Chedabucto Place, a subset would be grade 5 class.” • “Sets can have subsets with nothing in it. Cycles keep repeating.” • “We did a cycle and it was when we got in a circle and we threw the balls. The whole point was that a cycle continues and never changes.” • “A cycle can be from one thing back to that same thing.” • “ex: A set is a class. A subset is a group of people that play hockey in the class.” Exam Question Exam Question What are two things to assume about sets? • A set S is made up of elements, and if a is one of these elements, we shall denote this by a S. • There is exactly one set with no elements. It is the empty set and is denoted by . . • We may describe a set either by giving a property that characterizes the elements, or by listing all the elements. • A set is well defined. If S is a set and a is some object, then either a is in S, or a is not in S (definite) References • A First Course In Abstract Algebra by John. B. Fraleigh, Seventh Edition. • Grade 5 Class at Chedabucto Place.