maths Flashcards
what are natural numbers
N: positive integers
what are integers
Z: whole numbers
what are real numbers
R: numbers with no imaginary parts
what are complex numbers
C: numbers with real and imaginary parts
what are rational numbers
Q: numbers that can be written as fractions
what is exclusive and inclusive counting?
there are n up to and including / excluding this one
how do programming languages represent numbers?
they have a largest number and once exceeded, go back to the lowest number. more of a number ring
what does commutative and associative mean?
(using multiplication as an example but works for addition and more algorithms)
commutative: ab = ba
associative: a(bc) = (ab)c
what is a neutral and annihilating element?
an element that when a number and it are put together in the function return the original number and 0 respectively
what is a commutative ring and a commutative semiring?
they are both sets of numbers that where there are can be multiplied and added. A ring has an additive inverse meaning every number has another number that when added together returns 0. an example of a commutative ring and semiring would be integers and natural numbers respectively
what does [-3, 5) mean?
{-3, -2, -1, 0, 1, 2, 3, 4}
what does Σ{…} and Π{…} mean?
sum and product of the elements in the list
how do you find the negative version of a number in complement notation?
flip the bits and add one. for non binary number systems such as denary, flipping 0 gets 9, 1 gets 8 etc`
what does it mean for two numbers to be coprime?
they have no common factors
what is a field?
a commutative ring with a multiplicative inverse for ever number such that a * MI = 1
such as rational and real numbers
what laws are true if a number system has additive and multiplicative inverses?
additive cancellation law: if x+a = y+a, x=y
multiplicative cancellation law: if xa = ya, x=y
these laws
how do you calculate a floating point binary number?
mantissa: first digit is 1s column (decimal point is between 1st and 2nd digits)
exponent: whatever the number is, subtract the value of the largest bit
then: answer = m * 2^e
how do you work with numbers at a higher accuracy than the floating point can represent?
split the number up like 3.404 ×10^8 + 7.191 ×10^2 = 3.404007191 ×10^8
what does congruent modulo mean and how is it written?
if two numbers have the same remainder when divided by a number, they are congruent modulo (mod x)
eg: 168 ≡ 3468 (mod 10)
why is Zₘ always a commutative ring but Z isn’t?
because the additive inverse of a positive inverse is also positive because Zₘ = [0,m)
eg: m=10
3 +₁₀ 7 = (3+7) MOD 10 = 0 so 7 is the additive inverse to 3
when is Zₘ a field?
when every number in Zₘ apart from 0 has a multiplicative inverse meaning they all have to be coprime with m. This means m has to be prime
when are sets equal?
when they both include the same unique numbers no matter of the order or how many times each is included
how is the set of things in A that are not in B represented?
A\B
how can set builder notation be used to show
‘the set of all natural numbers whose cube is greater than or equal to 100’
{ n∈N| n³ ≥ 100}
what is the product of sets?
a single set of every combination of ordered tuples each containing a value from each set
(not associative)
If A is a set, what is A* and Aʷ?
A* is the set of all lists made of things in the set A such as if A is characters, A* is strings
Aʷ is the set of all infinite sequences (streams) of things in the set A such as if A is N, the numbers after the decimal point in pi would be a member of Aʷ
what is a powerset?
P(A) is the set of all subsets of A
what are the de morgan laws?
¬(A∧B) = ¬A ∨ ¬B
¬(A∨ B) = ¬A ∧ ¬B
what is a relation?
a set of ordered pairs that contains every item in set A (domain) that relates to an item in set B (codomain). this is also a subset of AxB. can also have relations of more than 2 sets
what is image and preimage?
for any relation R: A->B, b is the image of a and a is the preimage of b`
what does it mean for a relation to be single-valued?
each preimage only has one image
either one-to-one or many-to-one
what does it mean for a relation to be total?
every possible combination of elements are related either both ways or one way round
what is the empty and universal relations?
no elements are related to each other and R = AxB
what is a function?
a function is a relation where every item in the domain has a unique image in the codomain
if there is a function f: A -> B, and sets A and B have size n and m respectively, how many possible functions could be made for f?
m^n functions
how is the set of all possible functions from A to B represented?
B^A
what is an endorelation and an endofunction?
a relation/ function where the domain and codomain are the same set
what is a converse relation?
a relation that swaps the domain and codomain of the original relation (inverse)
what does it mean for a function to be injective?
the function is one-to-one, no image shares the same preimage (domain ≤ codomain)
what does it mean for a function to be surjective?
every item in the codomain has a preimage meaning the range is the codomain (domain ≥ codomain)
what does it mean for a function to be bijective?
when a function is both injective and surjective (both sets are the same size)
what is a reflexive and irreflexive relation?
reflexive relation: an endorelation when every item in the domain relates to itself
irreflexive relation: no item relates to itself
what is a symmetric and antisymmetric relation?
symmetric relation: an endorelation where if xRy, yRx
antisymmetric relation: distinct elements cannot be mutually related
what is a transitive relation?
an endorelation where if xRy and yRz, then xRz
what is an equivalence relation?
a relation that is reflexive, symmetric and transitive such as modularity, same age etc
what is true of a set if there is an equivalence relation?
a partition of the set is made which is when every member of the domain is also a member of a disjoint (non-overlapping) subset called an equivalence classes
how do you write equivalence classes and the set of them for a relation (mod 10) on natural numbers?
Z / ≡(mod 10) = { [0]₌ , [1]₌ , [2]₌ , [3]₌ , [4]₌ , [5]₌ , [6]₌ , [7]₌ , [8]₌ , [9]₌ }
what is the kernel of a function?
the equivalence relation on the set of the domain that groups them into equivalence classes that have the same image. the kernel has a one-to-one correspondence to the range of the function
what is an order?
an endorelation on a set that is reflexive, ANTISYMMETRIC, and transitive such as ≥ on N or ⊆ on PX
what is a linear order?
when any two elements (x,y) are comparable meaning that at least one of x=y, xRy, yRx is true. an example of this is ≥ on N
what is a gaussian elimination?
a way computers solve linear equations by taking them from the form ax+by+…=c into
[a b|c] and putting them on top of each other. then they can be reordered, multiplied my a constant or added to each other until solved.
what is a vector (no real answer on the other side)
kinda hard to explain but must obey the 8 rules (look them up) but an example of it would be the vector space of n-dimensional co-ordinates over the field R or matrices. the 8 rules mean that vectors behave like the example given over their own field
what is a spanning set?
a set of vectors that spans the whole vector space
what is a span of vectors?
a set of all linear combinations that can be made from the original vectors
what does linearly independent mean?
if av+bu+…=0, a and b must both be 0
can be checked if none of the vectors are parallel and there is the same number of vectors as dimensions
what is a vector space?
A group of vectors such as Q^2 that obeys the 8 rules and has some form of vector addition and multiplication
how do you calculate the inner product of two vectors?
sum the products of the corresponding elements in the vectors
what does it mean for a basis to be orthogonal?
all inner products of vectors in the basis are equal to 0
when can matrices be added and multiplied?
added: two matrices must have the same dimensions
multiplied: number of columns in the first matrix must be the same as the number of rows in the second matrix
how do you:
1: add two matrices
2: multiply a matrix by a scalar
3: multiply two matrices
1: add corresponding elements
2: multiply every element by the scalar
3: each final element is the inner product of the row of the first matrix and the column of the second where the row and column cross at the final element
cᵢₖ = Σ(j:1→n) aᵢⱼ x bⱼₖ
what is a basis?
a spanning set of vectors where the vectors are linearly independent
