Sets and Venn Diagrams (Edexcel) - GCSE Maths

Summary

Highlights

• Learn how to denote sets using curly brackets and capital letters for easy reference. 🔤
• Creating Venn diagrams involves overlapping circles to highlight shared numbers between sets. 🔵
• The universal set helps include all necessary numbers, keeping it all inside a rectangle for clarity. 🔳
• Explore intersection, union, and complement operations with easy-to-understand symbols. 📏
• Follow along with example problems to cement your understanding while tackling exam-style questions effectively. 📈
• Gain confidence in solving probabilistic queries by counting numbers within different set sections. 🎲
• Step-up your skills with examples involving three intersecting sets and uncover deeper insights. 🎓

Key Takeaways

• Sets are collections of numbers, often represented with curly brackets. This video teaches how they form the basis of Venn diagrams. 🧩
• Venn diagrams visually represent relationships between sets, with overlapping areas showing shared elements. 🔄
• Understanding the union and intersection of sets is key for solving complex Venn diagram problems. ➕
• The universal set includes all possible elements in the scenario, with non-included ones outside the circles. 🌌
• Probability questions using Venn diagrams can ask you to calculate the likelihood of elements being in different set regions. 🎲
• Prepare for three-set Venn diagrams for more advanced scenarios, essential for Edexcel exams. 📚

Overview

Chapters

• 00:00 - 00:30: Introduction to Sets and Venn Diagrams This chapter introduces the concept of sets and Venn diagrams. It is tailored specifically for students whose exam board is Edexcel, although a subsequent video will cover other exam boards. The chapter begins by explaining that a set is a collection of things, usually numbers, setting the stage for further exploration into the topic.
• 00:30 - 01:30: Understanding Sets and Their Notation In this chapter, 'Understanding Sets and Their Notation', the concept of sets in mathematics is discussed. A set is described as a collection of distinct elements or numbers. In particular, the chapter provides an example of a set of positive integers less than or equal to 10. It is explained that each number within the set is referred to as a member or an element. Sets are often depicted as lists of numbers enclosed within curly brackets to clearly define them. Additionally, sets are typically assigned a name, represented by a capital letter, for easy reference. For instance, a particular set might be labeled as 'Set A'.
• 01:30 - 03:00: Creating Venn Diagrams with Sets The chapter discusses creating Venn diagrams using sets, focusing on two example sets: set A and set B. Set B includes numbers that are factors of 12: 1, 2, 3, 4, 6, and 12. It notes the overlap with set A, which consists of positive integers less than or equal to 10, leading to the common elements 1, 2, 3, 4, and 6. These shared numbers exemplify the intersection of the two sets in the Venn diagram.
• 03:00 - 04:30: Using Universal Sets in Venn Diagrams The chapter discusses using Venn diagrams to represent universal sets, focusing on the placement of numbers in overlapping circles. Numbers that fit the criteria of both sets A and B are placed in the overlap area. For instance, the numbers 1 to 10 belong to set A, and factors of 12 belong to set B. However, numbers outside these parameters, like the number 11, are not included in either set, illustrating how universal sets encompass numbers not belonging to specific subsets.
• 04:30 - 09:30: Examples of Filling Venn Diagrams In this chapter titled "Examples of Filling Venn Diagrams," the concept of representing numbers that do not belong to any given sets (like sets A and B in a Venn diagram) is discussed. The speaker explains that numbers such as 13, 14, 15, -4, 100, 0, 0.7, Pi, and 6.13, among others, are not included in sets A and B, and there are infinitely many such numbers. To represent this conceptively in a Venn diagram, a rectangle is drawn around the circles, labeled with a special symbol (a squiggly 'e'), to indicate the universal set containing elements not in A or B.
• 09:30 - 12:00: Understanding Intersection and Union This chapter explains the concept of a universal set in the context of Venn diagrams. The universal set is defined as the set of all possible numbers that should be considered in a specific Venn diagram. For example, in the described scenario, the universal set is the positive integers from 1 to 15. Thus, every number from this range should appear somewhere in the Venn diagram. The chapter details that numbers 11, 13, 14, and 15 are notably absent, indicating they do not belong to the specified set.
• 12:00 - 16:00: Applying Venn Diagrams to Probability The chapter explains how Venn diagrams can be used to understand probabilities and set relationships visually. It begins with an example of placing elements that are not part of specific sets but still within the universal set, specifically how to arrange these elements inside the rectangle but outside the circles representing the individual sets. It discusses a scenario using a universal set with numbers from 1 to 10 and how to apply this to a Venn diagram. The example illustrates looking for numbers that belong to set A within this universal set.
• 16:00 - 19:30: Working with Three Sets in Venn Diagrams The chapter explains how to work with three sets in Venn diagrams. It demonstrates the process of identifying common elements between the sets and placing them in the overlapping sections of the Venn diagram. For instance, set A and set B share the numbers three and eight, which are placed in the overlapping section between these two sets. Unique numbers to each set are placed in their respective non-overlapping sections. The example continues by examining numbers found only in set A without overlap with set B, like the numbers one and five, which are placed distinctly in set A's circle.
• 19:30 - 20:00: Conclusion and Further Resources This chapter concludes the discussion on set theory, emphasizing the importance of carefully examining the universal set to ensure all elements are accounted for in sets A and B. It highlights a problem-solving strategy where numbers missing from both sets but present in the universal set are identified and placed correctly. This approach involves checking which numbers are exclusive to sets A and B and assigning them accordingly. The chapter also encourages further exploration and resources to solidify understanding of these concepts.

Sets and Venn Diagrams (Edexcel) - GCSE Maths Transcription

• 00:00 - 00:30 [Music] in this video we're going to learn about sets and vend diagrams unfortunately this is a topic that exam boards assess differently so this video is specifically for students whose exam board is at Excel don't worry though if your exam board is not at Excel because my next video will cover the other exam boards in maths a set is a collection of things which are usually numbers for
• 00:30 - 01:00 example this is the set of positive integers that are less than or equal to 10 each of the numbers within the set is known as a member or an element of the set we often write sets out as a list of numbers like this to make it clear that this is a set we write them inside curly brackets it's common to give the set a name which is a capital letter so we know the set we're talking about for example this could be set a let's have a look at a second set set
• 01:00 - 01:30 B and set B is going to be the set of numbers that are factors of 12 so all of the numbers that divide into 12 which are 1 2 3 4 6 and 12 so set B would look like this notice how there are some numbers that are in both set a and set B for example 1 2 3 4 and 6 this is because they're both positive integers less than or equal to 10 and also factors of 12 this means we're able to
• 01:30 - 02:00 overlap these circles to create a vend diagram where the numbers that are in both set a and set B are placed in this overlapping section in the middle notice how all of the numbers between 1 and 10 are still inside the a circle and all of the factors of 12 are still inside the B Circle but what about the numbers that are not inside either of the circles for example the number 11 well the number 11 is not a positive integer less than or equal to 10 so it's not in set a and it's not a factor of 12 so it's not in set B
• 02:00 - 02:30 so this would need to go outside of the circles but so with the number 13 and so would 14 and 15 and so would actually -4 and 100 or 0 or 0.7 or Pi or 6.13 in fact there are infinitely many numbers that are not inside the sets A and B to save us having to write down infinitely many numbers outside of these circles we draw a rectangle around the ven diagram and we label it with this squiggly e symbol here which stands for
• 02:30 - 03:00 the universal set the universal set is the set of all possible numbers that need to go inside the vend diagram let's say for example the universal set for this vend diagram is the positive integers from 1 to 15 that means all of the numbers from 1 to 15 must go inside this ven diagram somewhere now at the moment we have quite a lot of these numbers we have 1 2 3 4 5 6 7 8 9 10 and 12 that means we're missing 11 13 14 and 15 and that's because they're not in set
• 03:00 - 03:30 a and they're not in set B but they are still inside the universal set so they need to go inside the rectangle so what we do is place them inside the rectangle for the universal set but not inside the circles so somewhere like this this is now a completed vend diagram for these sets let's have a look at a second example imagine we had the universal set from 1 to 10 and we had set a and set B and we were asked to put this into a v diagram the first thing I would do is look for number that are inside set a
• 03:30 - 04:00 and set B since they need to go in the overlapping section in the middle I can see the number three is in both of those sets so I'm going to cross it out and place it inside the middle this is also true of the number eight so we'll cross that out and place it inside the middle now that's all of the numbers that were in set a and set B so let's look at just set a we have the number one inside set a but that's not inside set B so it will just go inside the left side of this a circle we also have the same for number five so that goes here as well then for set B we have the
• 04:00 - 04:30 number two but that's in B but not in a so it'll go in the right side of the B Circle here and the same for the number seven and the number 10 now we need to go back to the universal set and look for any numbers that we haven't included so far we've done 1 2 3 5 7 8 and 10 so we're missing the numbers 4 6 and 9 so these numbers are inside the universal set but not the sets A and B so they go inside the
• 04:30 - 05:00 rectangle but not inside the circles so we've got four 6 and 9 so this would be the completed ven diagram here are two questions for you to try just like the previous ones you have some sets and you need to write them inside the ven diagrams for the first question the number three is in both of the sets so that goes in the middle and so is the number five the number four is just inside set a so that goes on the left and so there's the number six the number eight
• 05:00 - 05:30 is only in set B so that goes on the right and the same for the number nine so far we've put the numbers 3 4 5 6 8 and 9 inside the ven diagram so we're missing 1 2 7 and 10 so outside of the circles but still inside the rectangle go the numbers 1 2 7 and 10 and that's your first completed vend diagram for the second question we can see that the number five is in both of those sets so that goes in the middle but that's actually the only number
• 05:30 - 06:00 next if we look at the numbers that are only in set C we have 1 2 4 6 and n and then numbers that are only in D which are 3 7 and 8 so far we've placed the numbers 1 2 3 4 5 6 7 8 and nine inside the ven diagram leaving us with 10 to go outside the circles but still inside the universal set sometimes a question could Define a
• 06:00 - 06:30 set in terms of words rather than numbers for example set a could be the set of odd numbers set B could be even numbers C could be prime numbers and D could be square numbers e could be factors of 20 and F could be multiples of five when we have sets that are written like this it's usually easier to write the set out in terms of numbers for example if we assume that all of our Universal sets do not include negative numbers then the set a of odd numbers would be 1 3 5 7 9 and so on
• 06:30 - 07:00 set B which is even numbers would be 0 2 4 6 8 and so on set C the prime numbers would be 2 3 5 7 11 and so on set D the square numbers would be 1 4 9 16 25 and so on set e the factors of 20 would be 1 2 4 5 10 and 20 and set F the multiples of five would be 5 10 15 20 and so on let's have a look at how we might use these in a question so here we have a
• 07:00 - 07:30 ven diagram and we have the universal set from one all the way to 20 set a is going to be even numbers and set B is going to be square numbers so I'm going to replace set a with a list of numbers that are even numbers but I'm only going to use the numbers in the Universal set from 1 to 20 so the even numbers between 1 and 20 are 2 4 6 8 10 12 14 16 18 and 20 then we'll do the same for set B which is the square numbers that are between 1 and 20 so 1 4 9 and 16 so all
• 07:30 - 08:00 we do in this question is rather than doing a for even numbers we use this set and rather than b for square numbers we use this set and then we just do like we did in the previous questions we'll look for numbers in both sets and I can see we've got four so that goes in the middle and we've also got 16 so that goes in the middle then we have all of the numbers that are just in set a so 2 6 8 10 12 14 18 and 20 and the two numbers that are just set b 1 and
• 08:00 - 08:30 9 then we need to check for any numbers that are still in the Universal set that we haven't used the numbers we have used so far are 1 2 4 6 8 9 10 12 14 16 18 and 20 so all of these remaining numbers need to go outside the circles but still inside the rectangle and they are three 5 7 11 13 15 17 and 19 so there's your complete V diagram here are two more
• 08:30 - 09:00 questions like this for you to try for the first question we'll need to write down the factors of six which are 1 2 3 and six we also need the factors of eight which are 1 2 4 and eight now we look for numbers in both of these sets and we've got one and two and they both go in the middle then for the numbers only in set a we have three and six and the numbers only in set B we have four 4 and 8 now we check which
• 09:00 - 09:30 numbers we've used in the Universal set which are 1 2 3 4 6 and 8 leaving us with five and seven to go outside the circles but still inside the rectangle for the second question we need to write down the odd numbers but only from 1 to 10 so they would be 1 3 5 7 and 9 and then we need the prime numbers from 1 to 10 so 2 3 5 and 7 the numbers that are in both C and D are 3
• 09:30 - 10:00 5 and seven so they go in the middle the numbers that are only in set C are 1 and 9 so they go on the left and the number two is the only number that's just in D so that goes on the right the numbers we've used so far are 1 2 3 5 7 and 9 leaving us with 4 6 8 and 10 to go outside of the circles but still inside the universal set next we're going to take a look at some of the regions on the vend diagram
• 10:00 - 10:30 you already know how to identify the set a it would be any of the numbers that are inside the circle which is labeled a the set B similarly will be any of the numbers inside the circle labeled B but what about this set here this symbol in between the A and B is known as an intersection symbol and we sometimes read it out loud as the word and so this set is a and b it's any of the numbers that are in the set a and also in the set B so this is another way of writing
• 10:30 - 11:00 down that overlapping section we talked about earlier this section here so we call this the intersection of A and B and we write it down with this symbol you also need to know about this set this symbol is the other way up and it's known as a union symbol we read this out loud using the word or so this is the set a or b is any of the numbers that are in the set a or the set B which is actually all of both of those circles
• 11:00 - 11:30 any of the numbers inside this region would be in set a or set B we also need to know about two more regions we need to know about a with a little Dash after it which we call the complement of a and sometimes we read this out loud as not a so it's all of the numbers that are not in the set a we already know the set a is represented by this circle here so not a would be any of the other regions on the ven diagram so all of this shaded section here so
• 11:30 - 12:00 the red region is the set not a in a similar way we can do B with a dash which is the complement of B which we would read as not B so if I shade the B Circle first of all the complement of B is everything that's not shaded so this red region here let's summarize the six regions we just looked at first of all we have the set a and then also the set not a which is the the opposite of set
• 12:00 - 12:30 a then we have B and the set not B which is the opposite of the set B then we had the intersection A and B which was just the overlapping section in the middle and finally we also had the union of A and B which we said was A or B which was all of both of those circles let's have a look at how these can be used in exam questions here we have a completed ven diagram and the question could say list all of the
• 12:30 - 13:00 numbers that are in set a set a is just the a circle so the numbers inside this set are 1 3 5 and 8 and in a very similar way set B would be all of the numbers inside the B Circle so 2 3 7 8 and 10 and what about if we were asked to list the numbers inside the set a-h or not a so here we need to identify all of the numbers that are not inside the a circle which are 2
• 13:00 - 13:30 4 6 7 9 and 10 in a similar way we can do the set not B which is all of the numbers not inside the B Circle so 1 4 5 6 and 9 we can also do the intersection of A and B so A and B which is the overlapping section in the middle so just three and8 and we can also do a union B which is a or b which is any of the numbers in e of those circles so 1 2 3 5 7 8 and 10
• 13:30 - 14:00 here's a question for you to try list all of the numbers in each of these sets so for set a we'd have any of the numbers inside the a circle so that's just 1 2 and four and for set not a any of the numbers not inside the a circle which is 3 5 6 7 8 and 9 for set B any of the numbers inside the B Circle so 2 3 5 6 and 7 and not B is any of the numbers not
• 14:00 - 14:30 inside the B Circle so 1 4 8 and 9 the intersection A and B is any of the numbers in the overlap so that's just two and then the union A or B is any of the numbers inside either of those circles so 1 2 3 4 5 6 and 7even we can also use these sets to answer questions relating to probability so take this ven diagram and the question says a number is chosen at random from the universal set find the probability that the number
• 14:30 - 15:00 is in set a so this time we're going to give the answer as a probability rather than a list of numbers the first thing I would do is count up how many numbers are inside the universal set since I'm choosing a number from that set this is the whole vend diagram you can see there are nine numbers here so the probability will be something out of nine then we need to look at the probability we've been asked for which is the probability the number selected is in set a so if we shade in set a we can see there are 1 2 3 four numbers
• 15:00 - 15:30 inside set a so the probability of selecting one of those at random will be four out of nine what if the probability was something different though like the set not B we do exactly the same idea but count up the numbers that are not in b not B looks like this so if we count up those numbers we've got 1 2 3 4 5 6 seven numbers that are in not B so the
• 15:30 - 16:00 probability will be seven out of n this time what if the probability that we were asked for was that it was in the set A and B well we shade in the set A and B which is the intersection in the middle and there's only one number there so the probability would be 1 over 9 and what if the probability Was A or B the union well this is the numbers inside either of those circles so we count them up and say there are 1 2 3 4 five numbers this time
• 16:00 - 16:30 so the probability is five out of N and we could even say what is the probability that the number selected is inside the universal set well the universal set is the entire diagram so that will be all nine numbers so it'll be nine out of nine and 9 out of n can be written as one so we'd give the answer one finally since edexel love you so much you need to be able to do everything you've done in this video with not just two sets but three sets so you need to be to coat with a vend diagram that looks something like
• 16:30 - 17:00 this let's have a look at a typical question where you might need to write numbers inside the vend diagram we have a universal set from 1 to 10 we have sets a b and c so this time I would start by trying to find any numbers that are in all three of those sets and I can see the number three is in a b and c that means it goes in the overlapping section of all three of the circles so right in the middle here there are no other numbers that are in sets a b and c
• 17:00 - 17:30 so I'll just look for numbers that are in sets A and B and I can see that four is in a and b so that goes in the section that's overlapping A and B but not overlapping C which is here I can then see that the number seven is in sets B and C but not in a so that goes in the overlap of B and C that isn't overlapping a which is here there are no other common numbers to a b and c so I've just got the numbers that are just in a just in B and just in C just in a
• 17:30 - 18:00 is 1 and two just in B is 9 and just in C is eight now I'll check with the universal set to see if there are any more numbers and we've used 1 2 3 4 7 8 and N9 but we haven't used five so that goes inside the universal set but not inside the circles and the same for six and the same for 10 now what about a follow-up question a number is chosen at random
• 18:00 - 18:30 from the universal set find the probability that the number is in the set a and b or a intersection B this time there are 10 numbers inside the universal set so the probability will be something out of 10 then we need to find the region A and B which is the overlap of A and B which is here there are two numbers here the number four and three so the probability will be two out of 10 what if the probability said B or C
• 18:30 - 19:00 well here we're looking for the entirety of the B and C circles so this region here and now we have 1 2 3 4 five numbers so the probability is five out of 10 and what if the probability said the set not a well that's anything that's not inside the a circle so we're looking at this region here and we've got 1 2 3 4 five six numbers so six out of 10 and what about if it said something
• 19:00 - 19:30 like this this means A and B and C so this is the overlapping region for all three circles right in the middle there's only one number here the number three so the probability will be one out of 10 and the question could also say A or B or C which would be the entirety of the a b and c circles so all of this region here if if we count up these we've got 1 2 3 4 5 6 seven numbers so
• 19:30 - 20:00 the probability is seven out of 10 thank you for watching this video I hope you found it useful check out the one I think you should watch next subscribe so you don't miss out on future videos and why not try the exam questions in this video's description