Logic, Sets, Methods of proof, Complex numbers, Functions

What's a proposition?

A proposition is a statement that is True or False

What’s a predicate?

A statement that depends on one or more variables and becomes a proposition when values are assigned to these variables. E.g. n is a prime number.

What do you use to denote predicates

P(n), P(n,m), Q(x) etc

What do we use to denote ‘or’?

We use V to denote ‘or’ e.g. P V Q means P or Q

What do we use to denote ‘and’

We use ∧ to denote ‘and’

What is the negation of a statement and what do we represent it by?

The negation of a statement P, ‘not P’ is true when P is false and vice versa.

What does is mean to be logically equivalent?

Two statements are said to be Logically Equivalent if they have the same truth tables.

If ‘P implies Q’, this is true in all cases except which one?

When P is true and Q is false.

If P is the hypothesis and Q is the conclusion, what can we say about Q when P is false

If P is false, then ‘P implies Q’ will always be true.

What is the contrapositive?

The contrapositive is “not Q implies not P”. This is logically equivalent to the original implication and hence if you prove one you prove the other.

What is the converse?

The converse of “P implies Q” is “Q implies P”

What is the negation of P implies Q?

P and not Q

What does P if and only if Q mean

Means P implies Q and Q implies P.

iff is the shorthand

What is a set?

A collection of objects. The objects in Set A are called the elements of A.

What is a proper subset?

Where two sets aren’t identical, with proper subsets having at least one fewer element than the larger set.

What does it mean for two sets to be disjoint?

A n B = the empty set, ie. No element from A is also in B

What is the difference of A and B (A\B)?

A but not B

What is the complement of A, A^c

Elements not in A

What is direct proof?

• Starts with some assumptions

• uses a chain of implications to reach a conculsion

• to prove P→Q is true, we assume P is true and then show that Q is also true

What is proof by contrapositive?

sometimes its easier to prove the contrapositive of the statement. This involves showing that if Q is false, then P must also be false, which is logically equivalent to proving P→Q. This is because they’re logically equivalent.

Steps for proof by contradiction

Assume the statement is false, i.e assume negation is true

Deduce from this assumption a statement we know is false or that contradicts one of our assumptions.

Conclude that the assumption is false, hence the original statement must be true.

Steps for simple mathematical induction proof

• Base Step: Verify the statement is true for the first value (usually n=1n = 1n=1).

• Inductive Hypothesis: Assume the statement is true for some arbitrary n=kn = kn=k.

• Inductive Step: Use the hypothesis to prove the statement is true for n=k+1n = k + 1n=k+1.

• Conclusion: By induction, the statement holds for all n≥1

What does i equal in complex numbers, and hence i²

i = √-1 so i²=-1

how do we do addition and subtraction with complex numbers?

Add/subtract the real and imaginary parts accordingly

How do we do multiplication with complex numbers?

Expand the brackets normally but this simplifies to:

(ac-bd) + i(ad+bc) for the complex numbers a+ib and c+id

Polar form: multiply the modulus’s and add the second argument to the first.

Exponential form: the same

What is the conjugate of a complex number?

For any complex number z=a+ib we define z̄ =a-ib

How do we do division with complex numbers

1. Multiply numerator and denominator by the conjugate of the denominator (c - di).

2. Simplify the numerator: (a + bi)(c - di) = (ac + bd) + (bc - ad)i.

3. Simplify the denominator: (c + di)(c - di) = c² + d².

4. Write in standard form: (a + bi)/(c + di) = (ac + bd)/(c² + d²) + (bc - ad)/(c² + d²)i.

Polar form:

Divide the magnitudes and subtract the second argument from the first.

Exponential form:

Divide the arguments and subtract the arguments.

How do we represent complex numbers?

To represent complex numbers we define the complex plane where z=x+iy is represented by the points (x,y). This plane is also called the Argard diagram.

What is the polar form of a complex number z?

z=r\left(\cos\theta+i\sin\theta\right) where r is the modulus of z and \theta is the argument of z.

r=+\sqrt{x^2+y^2}

What is DeMoivre’s Theorum for powers of a complex number?

Let z=r\left(\cos\theta+i\sin\theta\right)

z^{n}=r^{n}\left(\cos n\theta+i\sin n\theta\right)

How do you fin the roots of a complex number?

Let w^{n}=z for any natural number n

w=r^{\frac{1}{n}}\left(\cos\left(\frac{\theta+2\pi k}{n}\right)+i\sin\left(\frac{\theta+2\pi k}{n}\right)\right) for k is an integer.

What’s Euler’s formula?

e^{i\theta}=\cos\theta+i\sin\theta

What’s the exponential form of a complex number

z=re^{i\theta} Where r = |z| and theta=arg(z)

What is the fundamental theorem of Algebra?

Let n be contained in the natural number and let

a_0+a_1z+a_2z^2+\cdots+a_{n}z^{n} be a polynomial of degree n, wherea_0+a_1z+a_2z^2+\cdots+a_{n}z^{n} is contained in the complex numbers and a_{n}\ne0. Then this polynomial has exactly n complex roots, although some may be repeated.

What is the Cartesian product of a set?

Let A and B be non empty sets, then the Cartesian product of A and B (A x B) is the se of ordered pairs.

A\times B=\left\lbrace\left(a,b\right)\vert a\in A,b\in B\right\rbrace

We write A^2 For A x A

What is a relation?

A relation on a set A is any subset R of A×A. It shows which elements of A are related to which others. If (a,b) ∈ R, we say “a is related to b”. We write a~b for this.

What is a reflexive relation?

A relation R is reflexive if every element is related to itself: for all a ∈ A, (a,a) ∈ R.

What’s a symmetric relation

A relation R is symmetric if whenever (a,b) ∈ R, then (b,a) ∈ R too

What’s a transitive relation

A relation R is transitive if whenever (a,b) ∈ R and (b,c) ∈ R, then (a,c) ∈ R as well.

What’s an equivalence relation?

A relation that is reflex transitive and symmetric.

What condition has to be satisfied for a function.

It has to assign each element in A to a unique element in B

What’s the the image of a function?

The set of possible outcome values of said function. Also known as the range

What’s an injunctive function?

A function f:A→B is invective if

\forall x,y\in A,f(x)=f(y)\Rightarrow x=y

Also known as a one-to-one function.

What is a surjective function?

A function f:A→B is surjective if Imf=B.

\forall y\in B,\exists x\in A\vert y=f(x)

Also known as an onto function

What is a bijection?

A function that is injective and surjective.

If we have a bijection f:A→B we can say there is a one-to-one correspondence between A and B, meaning we can pair up the elements of A with the elements of B.