Lesson 17 Sets and subsets

# Episode 2

Reasoning Resources: Worksheet 2
Whole class preparation
Understanding that sets can overlap in such a way so that one is completely within another. Demonstrate what an included set is, by showing the multiples of 6 up to 54 in a set and then drawing an included set containing 12, 24, 36, 48. Ask for a description of the included set. Show an example of an excluded set, e.g. the set of multiples of 7 or 11.
Pair and small group work
Understanding that not all sets overlap at all, within the overall set.

Exploring the precise language of inclusion and exclusion.
Using Worksheet 2, pupils write down the set of numbers ‘multiples of 4’ from the numbers 1-40 listed at the top of the page. Can you find an included set? Ask them to try to find a set which is included within ‘multiples of 4. Remind the pupils to use pencils and rubbers.

A common mistake here is to choose a set which includes all of the original set, rather than a set which is included. For example, when looking for a set included in ‘multiples of 4’ they choose ‘multiples of 2’ rather than ‘multiples of 6,8, 10 or 12) etc.
Then ask pupils to write another set from the numbers 1-40 which is completely outside the set 'multiples of 4.' They should then write any remaining numbers from the 1-40 set outside all the rings.

Which is the ring with the fewest numbers in? Describe the numbers outside this ring. Describe the numbers outside all the rings but inside the overall set.
Whole class sharing and discussion
The number of elements in the union of two non-overlapping sets is the sum of their numbers, and number of a union of two sets one of which is included is the number of the inclusive set. Visually and in words this is much simpler than in Boolean algebra notation, e.g, n(AUB) = nA + nB — n(ANB) but the third term n(ANB) = 0 (AUB) =nA + nB.

Also if B is included, then n(ANB) = nB which cancels the middle term, so n(AUB) = HA.
Share results, emphasising that the included set cannot contain numbers which were not in the original set.

When describing sets of numbers in response to the questions above, show how different words and phrases can have the same meaning e.g.'except! ‘apart from’ and ‘not’ In the last question the language is likely to be ‘neither...nor! ‘apart from both’ and ‘except those in’. Explain that mathematicians like short clear phrases and ask which of those given come into this category.

Count the number of elements in each subset and add to get a total. Discuss why this is not the same as the number of elements in all the sets. The class may talk about sets that are impossible to show within the diagram, for example numbers that are not in the overall set, such as ‘three digit numbers’ or ‘fractional numbers.’