1. MA201 February 1 st, 2010 3.1 Presentation The Closure Property and Relating this to Properties of Sets Under an Operation

2. Presentation Outline A.The concept of closure is a challenging concept for elementary math teachers. - perhaps this is due to an unfamiliarity with sets, and a need for review - we skipped the book section on sets (2.1) B. I will go over the concept of closure using the ‘closed room analogy’, and then bring in some useful concepts about sets that can be used to formalize the concept of closure.

3. “Closed Room” Analogy Consider a set S = {0, 1, 2}. Is this set closed under addition? To answer this question, we imagine that the elements of S are placed in a closed room.

4. “Closed Room” Analogy Can sums be calculated from within the room? 0+0=? 0+1=? 0+2=? 1+1=? 1+2=? 2+2=?

5. “Closed Room” Analogy Can sums be calculated from within the room? 0+0=0 0+1=1 0+2=2 1+1=2 1+2=? 2+2=? Only some of the sums can be found within the room. This set if NOT CLOSED under addition.

6. “Closed Room” Analogy The ‘closure’ of the set under addition would need to contain 3 and 4. 0+0=0 0+1=1 0+2=2 1+1=2 1+2=3 2+2=4 The closure of S under addition would have to contain {0, 1, 2, 3, 4}. However, note that this new set is itself not closed under addition!

7. “Closed Room” Analogy Consider a set S = {0, 1, 2} again. Is this set closed under subtraction? As before, we imagine that the elements of S are placed in a closed room.

8. “Closed Room” Analogy Can subtraction be done from within the room? 0-0=? 0-1=? 2-0=? 0-2=? 2-1=? 1-0=? 2-2=? 1-1=? 1-2=?

9. “Closed Room” Analogy Can subtraction be done from within the room? 0-0=0 0-1=? 2-0=2 0-2=? 2-1=1 1-0=1 2-2=0 1-1=0 1-2=? Only some of the differences can be found within the room.

10. “Closed Room” Analogy Can subtraction be done from within the room? 0-0=0 0-1=? 2-0=2 0-2=? 2-1=1 1-0=1 2-2=0 1-1=0 1-2=? Exercise: What would the closure of S under subtraction have to contain?

11. Away from analogies: Definition of Closure Using Ideas About Sets A set S is closed under addition if you can perform addition on any two elements of the set and the sum is also contained within S. Generally: A set S is closed under an operation if you can perform that operation on elements of the set and the answer is always contained within S.

12. Away from analogies: Definition of Closure Using Ideas About Sets An equivalent definition: A set S is closed if it is equal to its closure. Two sets are equal if they have precisely the same elements. Two sets A and B are equal if every element of A is in B and vice versa.

13. Is the set of even numbers closed? E = {2, 4, 6, …} What would the closure of E have to contain? E is closed because it is equal to it’s closure: the sum of any two even numbers is even, which is the same set again!

14. Is the set of odd numbers closed? D = {1, 3, 5, …} What would the closure of D have to contain? D is not closed because the closure of D would have to contain even numbers as well, but these are not contained in D.

15. Subsets A set A is a subset of a set B if every element of A is also contained in B. –If B also contains more elements, than A is a proper subset of B and we say that A is ‘smaller’ than B. –{0, 2, 4} is a subset of {0, 1, 2, 3, 4, 5}

16. Group Problem #1 In groups of 5 or 6: A = {0, 5, 10, 15} Determine: what is the smallest set that would be the closure of A under addition? (That is find a set B that contains A and is closed under addition.)

17. Group Problem #2 All the sets we have seen that are closed under addition are infinite because they have an infinite number of elements. Examples: The set of whole numbers, the set of even numbers, the set of all multiples of 5. Can you think of a set that is closed under addition but that is not infinite?