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The prompt
The prompt
Mathematical inquiry processes: Interpret and reason; generate examples. Conceptual field of inquiry: Set notation; elements of sets; Venn diagrams.
The prompt shows De Morgan's Laws for complements in set theory. For any two sets A and B,
The complement of the union of the two sets is equal to the intersection of their complements and
The complement of the intersection of the two sets is equal to the union of their complements.
In the classroom, the inquiry starts with students pondering the meaning of the identities. They will pose questions about the symbols, noticing that the symbols for union and intersection have been 'swapped round'. As set notation is arbitrary knowledge (see the section below on Hewitt's distinction between arbitrary and necessary knowledge), the teacher has to be prepared to explain the meaning of the symbols at the beginning of the inquiry.
In a structured inquiry, the teacher could illustrate the laws by representing sets of integers in a Venn diagram (as in the example below). Students might then decide how to proceed by selecting a regulatory card. If they feel confident, they could represent their own sets in Venn diagrams and list the elements of the union, intersection and complements. As the prompt is true in all cases, students will be able to check the elements in their solutions. Otherwise, students might opt to work on sets provided by the teacher and limit themselves to thinking about the union and intersection before moving on to the complement.
The laws can also be shown to be true at a more general level by shading the regions of Venn diagrams.
Four lines of inquiry
Four lines of inquiry
- Shading regions
There are 16 different ways to shade the regions of a two-set Venn diagram. Can you describe each one using set notation? How many ways are there to shade the regions of a three-set Venn diagram (with intersecting sets)? Can you describe each one using set notation?
2. Proofs of the laws
2. Proofs of the laws
Look up the symbols used in set theory and mathematical proof to understand the proof above of one of De Morgan's laws. You can find both proofs here.
3. Three sets
3. Three sets
Use Venn diagrams to show that the following identities are true for any three sets A, B and C.
4. De Morgan's laws for set difference
4. De Morgan's laws for set difference
Use Venn diagrams to show that the following identities are true for any three sets A, B and C.
Arbitrary and necessary knowledge
Arbitrary and necessary knowledge
In his article Arbitrary and Necessary Part 1: a Way of Viewing the Mathematics Curriculum, Dave Hewitt distinguishes between arbitrary and necessary knowledge:
I describe something as arbitrary if someone could only come to know it to be true by being informed of it by some external means - whether by a teacher, a book, the internet, etc. If something is arbitrary, then it is arbitrary for all learners, and needs to be memorised to be known.... It is not only labels, symbols or names which are arbitrary. The mathematics curriculum is full of conventions, which are based on choices which have been made at some time in the past. For anyone learning those conventions today, they may seem arbitrary decisions....
There are aspects of the mathematics curriculum where students do not need to be informed. These are things which students can work out for themselves and know to be correct. They are parts of the mathematics curriculum which are not social conventions but rather are properties which can be worked out from what someone already knows.... So, the mathematical content which is on a curriculum can be divided up into those things which are arbitrary and those things which are necessary.
All students will need to be informed of the arbitrary. However, the necessary is dependent upon the awareness students already have.... Although [some knowledge] is necessary, it does not imply that all students have the awareness to be able to work [it] out, only that someone is able to work [it] out without the need to be informed.
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