# Logic, Sets & Probability

'A guide for for teaching! Ideas and resources and lessons to look forward to'

A fantastic element of the course, this unit gives importance to a part of Mathematics that is otherwise easily overlooked. It's lovely the way the three sub topics link together to create a global understanding. The process of logical deduction is an inherent part of the structure of so much knowledge and Mathematics in particular! As such, its a great opportunity to explore different contexts and bring in ToK.

This page is a guide to the topic, ideas for classroom activity that encourage critical thinking, practise exercises and some of the best resources available on the Internet to help! The overview written below aims to help teachers think about the main objectives of the module and potential opportunities and issues there are with teaching it! Please follow links to the pages on Ideas and resources, exercises and the internet guide.

#### Ideas and resources

Classroom experiences are the most important thing that we as teachers are responsible for. This links to a page of activities and ideas for teaching this unit that are aimed to be both engaging and effective in encourage students to enjoy, discover and understand mathematics. The list is a brief outline of an activity that links to a page that gives more detail and the associated resources. This is an area that should develop regularly. Think about subscribing to the RSS feed to get notifications of updates.

#### Practice

This page has a variety of tasks designed for practise and revision. Practice questions are an important part consolidating students' understanding of a concept. They can be found from numerous sources and the following is not intended as a 'fix all' solution. Most teachers appreciate that questions gathered from numerous different sources make the best diet!

#### The internet guide

This page is intended to list and outline some of the best resources available online to support the teaching and learning of this module. These will mostly be videos, virtual manipulatives and self help sites. Where appropriate, there will be a short commentary of what they are and how they might best be employed! It is also intended that this selection grow over time and that users contribute.

## Overview

I really enjoy the combination of topics in this unit. Most students will have seen this level of probability before, but putting it together with set theory and logic give us a real opportunity to explore the topic properly. Some students find Venn diagrams a much more intuitive means of understanding probability than tree diagrams so it is nice to use them side by side. My tendency is to teach the sub topics in the order the title suggests because they flow so nicely in that order. Sets and Logic is often a totally new idea to students so its important to spend some time exploring these ideas and the relationship between them. The variety of contexts available to explore these in make it fun as well.

The ideas are put to use straight away when looking at the probability. I round the unit off by looking at the laws of probability that are, of course, given using set notation! Above all, this topic rich in potential for looking at Mathematics that models our world and upon which so many decisions are made.

### Key Elements

- Understanding the concept of sets, compliments, subsets, unions and intersections and the corresponding set notation
- Constructing and Interpreting Venn Diagrams
- Understanding logic - the relationship between words and symbols
- Working with compound logical propositions and their corresponding truth tables
- Logical relationships, equivalence, tautology, contradiction
- Probability of single and combined independent events
- Symbolic laws of probability including conditional probability
- Problems linking all of the above ideas

### Teaching Time

The recommended time is 20 hours for this unit. It can be done, but it doesn't leave much time for exploration of some very interesting ideas. I will typically spend about 25 hours here, reclaiming some of the time by setting some statistics work during the year 12 holidays^{[1]}

### Sticking Points

This is a hard question to answer! All students are different, but the following is a list of things that come up most frequently

- Intersections (and) and unions (or), when shading in a union between two sets you shade in all of one set
**and**all of the other. This can cause some confusion - The total of the elements in three intersecting sets is not the sum of the three separate totals. This is counter intuitive in the beginning
- Going between logical expressions and symbols! Ands, ors, neithers, eithers and if thens! It is very easy to get lost in the words.
- Implication - why if both propositions are false is the compound implication True?
- Knowing when to apply the laws of probability. Often a problem looks approachable without them

### Teaching Opportunities

Sets and Venn diagrams can be really interesting in a variety of non-mathematical contexts as well as mathematical ones. For example a class could conduct a survey to see if students were pro death penalty, pro abortion and pro assisted suicide. This can represented in a Venn diagram and then analysed. Is anyone pro death penalty but anti-abortion? The diagram assist the debates in all of these issues. There are numerous other possibilities to tackle broader issues with the Mathematics. See the resources and ideas page.

Logic is in many ways at the very heart of Mathematics. Its logical deduction that proves theorems from axioms and studying Logic gives a great chance to explore the very nature of the Mathematics. Likewise the art of a well constructed argument can be analysed using logic. The study of logical fallacies has a strong link with ToK and adds interest to this unit, again, with or without Mathematical contexts. 'If all squares are rectangles, does that mean that all rectangles are squares?' These questions are great for linking the logic with the Venn diagrams. Link to Physics with and, or and not gates in electronic circuits.

Probability is a topic that has a huge impact on our society from gambling, to forecasting and reliability. As I have said before, studying the Sets and Logic before lends a nice perspective to probability and the possible contexts are many and varied. What is the probability of a given adult from the planet earth being an illiterate female? or of a false positive test for a particular disease? How does a fairground trader set up his games so that you can win, but you are most likely to lose? These questions and more...!

Link all three topics together as often as possible

## Selected Pages

### L, S & P Internet Guide 13 January 2018

This page is intended to list and outline some of the best resources available online to support the teaching and learning...

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### L,S&P Teaching Ideas 28 September 2016

Here you should find a wealth of resources that can be used by you and your students for teaching this topic. The resources...

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### Probability Trees 27 September 2016

'Use what you already to know to see just how and why probability tree diagrams work'Probability is a topic that moves very...

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### Dopey Logic 27 September 2016

'Can you make a sound and valid argument for the legalisation of Marijuana?'The 'legalise it' debate is one that is regularly...

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### Focus on Probability 1 June 2019

Possibly the most relevant, useful and appropriate part of any mathematics course in schools! Everyone has an intuitive...

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### L, S & P Practice 27 January 2019

This page has a variety of tasks designed for practise and revision. Practice questions are an important part consolidating...

more