A set is a very simple idea on the face of it, it is merely a collection or group; that is all. Sets are not objects in the real world; they are created within our minds not by our hands. An example of a set is all even numbers or all of the children in Europe or even the set of green bumblebees. But that is obviously not the sort of set we are interested in, we are only interested in ones that have interesting mathematical properties.
Sets by themselves may seem pretty uninteresting but when used in other areas of mathematics they show how powerful they truly are. They are used as a foundation from which the majority of mathematics can be derived, that is why they're so fundamentally important to mathematicians! Sets have to be defined in such a way that creates some list of numbers.
By convention, sets are denoted by capital letters, elements (read more in the next paragraph) are labelled by lower case letters. Beyond this the notation for sets is basic: you list each element, separated by a comma, and surround this list by curly braces, {}. For example, some set, Q, contains elements {p, q, r}.
There are a range of ways in which a set may be defined, either the items are defined using a semantic description for example: all of the odd integers or all prime numbers divisible by 4. Or you can define a set by listing the elements individually, for example: {2, 4, 8, 16}, with this version an ellipsis may be used to show that the set continues indefinitely in the same manner {3, 9, 27, 81, ...} like so.
A set constitutes of elements, an element is one distinct item that makes up a set. For example 1 is an element of the set of integers, the notation for an element, x, belonging to a set R is: x ∈ R. x ∉ R means x is not an element of R.
A subset is a set that entirely lies within another set, for example all prime numbers are integers. A ⊆ B means every element of A is also an element of B and this is read as "A is a subset of B". An interesting point from this definition is that for any set, S, every element of S is clearly also in S thus S is a subset of S (S ⊆ S). This is a bit strange so we introduce something more thorough than subsets and that is proper subsets: A is a proper subset of B if every element in A is also in B and there is at least one element in B that is not in A; this has the slightly different notation of A ⊂ B.
The next two you may be more familiar with if you have done maths past GCSE level you may have encountered them, they are union and intersection. A union B is essentially A and B, the notation for this is A ∪ B, it is everything that is in A and B. A intersection B is where A and B overlap and the notation for this is A ∩ B.
The union of all of the natural numbers {1, 2, 3, ...} and all of the integers {-2, -1, 0, 1, 2, ...} will result in just the set of integers, the reason for this is that it does not matter if a set contains duplicate identical elements it is the same as long as the same elements are present, not how many. {1, 2, 3} is the same set as {1, 1, 2, 3, 2, 3} as the same elements are present in both. It then follows that A ∪ B = A + B - A ∩ B, which is a vitally important fact for combining sets.
You may also be interested in the size of infinite sets, which investigates that there are some infinities larger than others!
Showing posts with label numbers. Show all posts
Showing posts with label numbers. Show all posts
Saturday, 6 October 2012
Monday, 31 October 2011
Different Sizes of Infinity
Before you read this make sure you have a basic knowledge of sets.
The "cardinality" of something is the size of a set. For example the cardinality of the set {3, 2, 17, -9} is 4, and the cardinality of the set {2, 6, -7} is 3. But when thinking it terms of infinity things become more confusing, I mean how can you categorise different sizes of infinity, infinity surely must just be infinity, right?
Well unfortunately, no. Using the ideas of sets we can find that there are actually different cardinalities of infinity. But to find this we need a more general definition of what being the "same sized set" is. A set {1, 2, 3, ..., x-1, x} has a cardinality of x and the set {1, 2, 3, ..., y-1, y} has a cardinality of y, the sets will be the same size if x = y. This then leads to the definition that being the "same size" is if the sets have the same amount of elements.
When we check if a set is the same size as another set, we simply count the number of elements in both sets and compare. But with infinite sets this is (obviously), impossible. But if we break the task down a bit we find ways of managing this. Take the elements from each set and pair them up, if they do match up perfectly with no doubling up or ignoring any elements then the sets are the same size.
Now if we pair the natural numbers, N {1, 2, 3, ...}, and the integers, Z {-2, -1, 0, 1, 2, ...}, we will be able to see which set is larger. We pair the numbers up like so:
And you can see, that despite your better knowledge, the numbers are pairing up and will continue to do perfectly infinitely. This then means that the sets of both the natural numbers and integers are the same size and thus have the same cardinality. This cardinality is referred to as: ℵ0 (read as "aleph null" or "aleph zero"). The aleph, "ℵ", character is used as it is the first letter in the Hebrew word 'infinity'. ℵ0 refers to 'countable infinity', or in simpler terms the lowest possible form of infinity.
When we continue to pair up the numbers we discover that the rational numbers and the natural numbers are too the exact same size and thus the cardinality of the rational numbers is also ℵ0. In fact the prime numbers are also deemed to have the same cardinality as the natural numbers, integers and rational numbers. I know it seems insane, but it is true.
And going back to my earlier point there are larger cardinalities of infinity. As you would think the next cardinality of infinity is ℵ1. The size of the real numbers in fact are larger that the natural numbers, integers, etc., this is because the real numbers (in now way at all) can be paired to the natural numbers. Because, between 0 and 1 there an infinite number of real values that are there, in fact the same can be said between 0 and 0.000000001. Thinking of it in this way helps to visualise the fact that the set of real numbers must be larger than the set of natural numbers.
As far as ℵx goes it will exist as long as x is a natural number, but as x goes past 1 it becomes much harder to visualise how this infinity is larger than the last. But there can not be ℵ1.5 for example, this is the continuum hypothesis.
The "cardinality" of something is the size of a set. For example the cardinality of the set {3, 2, 17, -9} is 4, and the cardinality of the set {2, 6, -7} is 3. But when thinking it terms of infinity things become more confusing, I mean how can you categorise different sizes of infinity, infinity surely must just be infinity, right?
Well unfortunately, no. Using the ideas of sets we can find that there are actually different cardinalities of infinity. But to find this we need a more general definition of what being the "same sized set" is. A set {1, 2, 3, ..., x-1, x} has a cardinality of x and the set {1, 2, 3, ..., y-1, y} has a cardinality of y, the sets will be the same size if x = y. This then leads to the definition that being the "same size" is if the sets have the same amount of elements.
When we check if a set is the same size as another set, we simply count the number of elements in both sets and compare. But with infinite sets this is (obviously), impossible. But if we break the task down a bit we find ways of managing this. Take the elements from each set and pair them up, if they do match up perfectly with no doubling up or ignoring any elements then the sets are the same size.
Now if we pair the natural numbers, N {1, 2, 3, ...}, and the integers, Z {-2, -1, 0, 1, 2, ...}, we will be able to see which set is larger. We pair the numbers up like so:
And you can see, that despite your better knowledge, the numbers are pairing up and will continue to do perfectly infinitely. This then means that the sets of both the natural numbers and integers are the same size and thus have the same cardinality. This cardinality is referred to as: ℵ0 (read as "aleph null" or "aleph zero"). The aleph, "ℵ", character is used as it is the first letter in the Hebrew word 'infinity'. ℵ0 refers to 'countable infinity', or in simpler terms the lowest possible form of infinity.
When we continue to pair up the numbers we discover that the rational numbers and the natural numbers are too the exact same size and thus the cardinality of the rational numbers is also ℵ0. In fact the prime numbers are also deemed to have the same cardinality as the natural numbers, integers and rational numbers. I know it seems insane, but it is true.
And going back to my earlier point there are larger cardinalities of infinity. As you would think the next cardinality of infinity is ℵ1. The size of the real numbers in fact are larger that the natural numbers, integers, etc., this is because the real numbers (in now way at all) can be paired to the natural numbers. Because, between 0 and 1 there an infinite number of real values that are there, in fact the same can be said between 0 and 0.000000001. Thinking of it in this way helps to visualise the fact that the set of real numbers must be larger than the set of natural numbers.
As far as ℵx goes it will exist as long as x is a natural number, but as x goes past 1 it becomes much harder to visualise how this infinity is larger than the last. But there can not be ℵ1.5 for example, this is the continuum hypothesis.
Saturday, 29 October 2011
Modulus
You may or may not be aware of what modulus is. If you have come across the modulus (absolute value) of a number before it is likely to only be of real numbers. The notation for the modulus of x is, |x|.
If the number is just a real number then calculating the modulus is incredibly easy and requires no thought. Essentially it is just 'taking the positive value' of the number, for example: |5| is simply 5, and |-3| = 3. It follows that any rational or irrational real number is just the positive value of it. The definition of |x| for any real value of x therefore follows that |x| = √(x²).
Another, potentially easier way to think of the modulus of a number, is to think of the numbers distance from 0 on a number line. -5 is 5 units away from 0 and 5 is also 5 units away from 0. This thought is useful when it comes to finding the modulus of a complex number.
Complex numbers can also be represented (in a way) on a number line, they are represented in a two-dimensional complex plane. A complex number, 5+3i will be 5 units in the x direction and 3 units in the y direction, therefore the general form of a complex number is x+iy.
As I said previously, to find the modulus of a complex number it helps if we think of our previous definition of |x| being the distance from 0 on a number line (or a complex plane, in this case). If we think of it in this manner then simple co-ordinate geometry states that the distance from 0 to x+iy is √[(x-0)²+(y-0)²], or simply the general definition of |x+iy| = √(x²+y²).
|x| = √(x²) when x is a real number, and this able to be proved from our new definition of |x+iy|. |x+iy| = √(x²+y²), and when x+iy is a real number, y has to equal 0, plugging this into the equation we get |x+i0| = √(x²+0²), therefore |x| = √(x²), proved.
If any of this was explained poorly or just went straight over your head please comment and I will do everything I can to make it understandable. Also if you would like more information on imaginary numbers please visit an older post of mine here.
If the number is just a real number then calculating the modulus is incredibly easy and requires no thought. Essentially it is just 'taking the positive value' of the number, for example: |5| is simply 5, and |-3| = 3. It follows that any rational or irrational real number is just the positive value of it. The definition of |x| for any real value of x therefore follows that |x| = √(x²).
Another, potentially easier way to think of the modulus of a number, is to think of the numbers distance from 0 on a number line. -5 is 5 units away from 0 and 5 is also 5 units away from 0. This thought is useful when it comes to finding the modulus of a complex number.
Complex numbers can also be represented (in a way) on a number line, they are represented in a two-dimensional complex plane. A complex number, 5+3i will be 5 units in the x direction and 3 units in the y direction, therefore the general form of a complex number is x+iy.
As I said previously, to find the modulus of a complex number it helps if we think of our previous definition of |x| being the distance from 0 on a number line (or a complex plane, in this case). If we think of it in this manner then simple co-ordinate geometry states that the distance from 0 to x+iy is √[(x-0)²+(y-0)²], or simply the general definition of |x+iy| = √(x²+y²).
|x| = √(x²) when x is a real number, and this able to be proved from our new definition of |x+iy|. |x+iy| = √(x²+y²), and when x+iy is a real number, y has to equal 0, plugging this into the equation we get |x+i0| = √(x²+0²), therefore |x| = √(x²), proved.
If any of this was explained poorly or just went straight over your head please comment and I will do everything I can to make it understandable. Also if you would like more information on imaginary numbers please visit an older post of mine here.
Sunday, 13 March 2011
Googol
No, my title is not a spelling error of Google. Googol is a number, a big number, a number that was so large and intriguing it inspired two nerdy University Student to create an easily available to everyone search engine to bring all of the internet to anyone. But talking about Google being awesome is a different article for a different time.
In this article I'm going to discuss the number a googol and a googolplex. Firstly what are they? Well a googol is 10100, that is a one followed by one hundred zeros, or the less catchy title, ten duotrigintillion. Just how big is that then? Well, there is estimated to be about 1.33×1050 atoms in the entire Earth. A googol is almost 1050 as much as this number. Nothing like maths to make you feel tiny and insignificant.
If you think that's large, your head may just explode when you're introduce to a googolplex. A googolplex is 10googol, that is enormous. A one followed by a googol zeros. That is so large it is almost impossible to comprehend. I'll try my best to explain just how mammothly big this number is.
Astronomer Carl Sagan estimated that writing a googolplex would be impossible, why? It would require more space than the known universe can provide. Seriously. There is only about 2.5×1089 elementary particles in the known universe, so even if every particle was a digit, it would still need the observable universe to be about one hundred billion times larger.
But, if you could fit the number into the known universe it would take an unbelievable amount of time to write. If you could write 120 digits a minute, it would take around 1.51×1092 years to finish (almost a googolplex years!). This amount of time is 1082 times the age of the universe. So, I wouldn't really start trying to write it out. There's a high chance, that even if it was possible to write for that period of time, that the entire universe will have inevitably died.
A more nerdy way to try and explain the size of a googolplex is to look at Planck time. Planck time (which I won't go into too much detail now), is the time it takes for light to travel one Planck length in a vaccuum, which is roughly 5.4×10-44, this is unbelievably quick. A googol is larger than the estimated amount of Planck times that have past since the start of the universe (8×1060), so a googolplex is much, much larger than even the smallest measurement of time that has past since the dawn of everything.
There is however a larger number that still has a use, it is known as Graham's number. But that's a different article for a different time. I hope this post has given you a small idea into the enormity Maths can hold.
In this article I'm going to discuss the number a googol and a googolplex. Firstly what are they? Well a googol is 10100, that is a one followed by one hundred zeros, or the less catchy title, ten duotrigintillion. Just how big is that then? Well, there is estimated to be about 1.33×1050 atoms in the entire Earth. A googol is almost 1050 as much as this number. Nothing like maths to make you feel tiny and insignificant.
If you think that's large, your head may just explode when you're introduce to a googolplex. A googolplex is 10googol, that is enormous. A one followed by a googol zeros. That is so large it is almost impossible to comprehend. I'll try my best to explain just how mammothly big this number is.
Astronomer Carl Sagan estimated that writing a googolplex would be impossible, why? It would require more space than the known universe can provide. Seriously. There is only about 2.5×1089 elementary particles in the known universe, so even if every particle was a digit, it would still need the observable universe to be about one hundred billion times larger.
But, if you could fit the number into the known universe it would take an unbelievable amount of time to write. If you could write 120 digits a minute, it would take around 1.51×1092 years to finish (almost a googolplex years!). This amount of time is 1082 times the age of the universe. So, I wouldn't really start trying to write it out. There's a high chance, that even if it was possible to write for that period of time, that the entire universe will have inevitably died.
A more nerdy way to try and explain the size of a googolplex is to look at Planck time. Planck time (which I won't go into too much detail now), is the time it takes for light to travel one Planck length in a vaccuum, which is roughly 5.4×10-44, this is unbelievably quick. A googol is larger than the estimated amount of Planck times that have past since the start of the universe (8×1060), so a googolplex is much, much larger than even the smallest measurement of time that has past since the dawn of everything.
There is however a larger number that still has a use, it is known as Graham's number. But that's a different article for a different time. I hope this post has given you a small idea into the enormity Maths can hold.
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