Maths - Semester 2 Flashcards
(18 cards)
What are the two definitions of a permutation?
A bijective function from a set to itself. where each set is mapped to exactly one other element inside the set.
It is also defines as the arrangement/ordering of elements in a set of objects.
How are the two definitions of permutations related?
A permutation bijective mapping places elements from a set into an identical set with identical elements, simply in a different order, as each element has one and only one counterpart in the set through the permutation. As a result, we can see that a permutation orders sets.
What is the set of all possible permutations, how many permutations will it have, and why does this make sense?
The set of all possible permutations is a set containing every single permutation function that can work on another set, denoted S_n.
The number of permutations in a set S_n is given as n!, which makes sense as a permutation is one ordering of numbers 1 to n in a set, and so the number of permutations is the number of ways of ordering numbers 1 to n, which is just n!.
What is the composition of permutations f and g of set S?
The composition of permutations f and g is simply another permutation of the set S
How does permutation multiplication work?
Permutation multiplication is the act of composing specific permutation functions to result in others within the set.
How do powers of permutations work?
Permutation powers work when a permutation is composed with itself, and is given a power notation for simplicity.
What is the identity permutation?
The identity permutation is a permutation such that all elements are ordered into their current position, and so essentially do not move.
What is an inverse permutation?
An inverse permutation is one that when composed with a permutation, results in the elements not moving (or in other words gives the identity permutation).
What are the 4 rules of permutation composition?
- If f and g are permutations, then so is their composition.
- For any f,g,h in a set, f(gh)=(fg)h, which means that compositions of 3 sets result in the same output no matter which is composed first
- The identity permutation is that in which every element is mapped to itself
- Every permutation has its own inverse
How is cycle notation used to write permutations?
It is done by listing each number which directly connects with the next in the new set within its own bracket, and placing all such brackets next to eachother:
f = (14367)(25)(8)
What are disjoint cycles?
Disjoint cycles are those which don’t have elements in common. If a permutation did not have a disjoint cycle, it would simply be one big bracket where every number is linked
What is the order of permutations, and how is it found for a permutation of more than 1 cycle?
The order of permutations is the smallest number of times in which a permutation is repeated until the original set is reached. For a single cycle permutation, it is simply found as the number of elements in the cycle.
For a multi-cycle permutation, the lowest common multiple of the size of such cycles is instead the answer.
When are even and odd permutations applicable, and what are they?
Even and odd permutations indicate the number of element swaps it takes to make a permutation. An even permutation is one which takes an even number of swaps, and an odd is one which takes an odd number of swaps.
What is the quick formula for the signature of any r-cycle, and for permutations of multiple r-cycles?
1 r-cycle = (-1)^(r-1)
many r-cycles = =(−1)^(r_1−1) (−1)^(r_2−1)…(−1)^(r_n−1)
How is the area between two curves found?
It is simply the integral of the differences between the curves integral( f(x) - g(x))
How is the volume of a 3d object found?
Integrating an initial curve representing its circumference to find the area under the curve, and then integrating this area
What is the solid of revolution and how is it found?
The solid of revolution is simply a volume of space, and is found by rotating a region around a line, which if the line is encompassed by the curve is given by integrating pi*r^2, and if a disk where the axis of rotation is not entirely covered, is given by integrating πR^2−πr^2.
How do you find the volume of revolution in a way which involves slicing parallel to the axis of rotation?
This is done by integrating a function consisting of a circumference (an infintely small band of the volume in the x-axis), multiplied by the height of the function (given by f(x))
