# Set 7 Flashcards

What is important about the components of a vector?

That they are all drawn from the same **field** of values, e.g Reals or Rationals

Give four ways vectors be represented

- A list of values
- A one-dimensional array
- A dictionary (if the vector can be considered to represent a function)
- Visually (using an arrow)

How can a vector be represented in code if the vector is viewed as a function?

A dictionary

What does vector addition achieve?

Translation

What does scalar-vector multiplication achieve?

Scaling

What is another name for dot product?

Scalar product

What is a convex combination of vectors **u** and **v**?

α**u** + β**v**

such that α , β >= 0

and α + β = 1

What is the scalar product of vectors: A = (a1,a2) and B = (b1,b2)

A.B = a1b1 + a2b2

What is an application of the dot product?

Finding the angle between two vectors

In a convex combination of **u** and **v**, what is the significance of α = β = 0.5?

The resultant vector is the **midpoint** of the *position* vectors **u** and **v**

When might it be appropriate to represent a vector using a dictionary?

If the vector is used to represent a **function**

What is a set?

A set is an **unordered** collection of values in which each value occurs **at most once**

What does proper subset mean? What is its symbol?

- Equality not allowed
- (a set is a subset of itself but
*not*a proper subset of itself) - ⊂

A\B

Set difference between A and B

What is a finite set?

One whose elements can be counted off by natural numbers up to a particular number

What is the Cartesian product of A and B?

The set of all ordered pairs (a, b) where a is a member of A and b is a member of B.

What can the Cartesian product be used for?

Defining function types

What is a countable set?

A set with the same cardinality as some subset of the natural numbers

What are regular expressions?

Metalanguage for defining all valid strings of a regular language

What is the purpose of regular expressions?

Allow for regular languages to be described in a convenient shorthand notation

What is the RegEx metacharacter for: zero or more

*

What is the RegEx metacharacter for: one or more

+

RegEx : zero or one lots of

?

What are | ? * + called in RegEx?

Metacharacters

Context free languages are a __________ of regular languages

Superset

Why are context-free languages needed? (2*)

- RegEx doesn’t have enough
**expressive power** - This is because regular languages can be expressed in terms of a finite state machine, and a finite state machine only has finitely many states

What is BNF?

- Backus-Naur form
- A metalanguage for describing the syntax used by a context-free language

General statement in BNF

LHS ::= RHS

What is a function?

- A function is a mapping of values from a domain to a set of values from a co-domain.
- Not all of the co-domains members needs to be outputs.

Notation for *“the set difference between sets A and B”*

A\B

Set comprehension for all even natural numbers less than 10

{2x | x ∈ ℕ ∧ x<5}

(| means such that, ∧ means AND)

What sorts of strings match {0ⁿ1ⁿ| n ≥ 1}

{ 01, 0011, 000111, 00001111, … }

How can you identify normalised floating point numbers?

Mantissa must begin with 01 or 10

Give two advantages of normalisation of floating point numbers

- It maximises precision for a given number of bits
- It ensures that there is a unique representation of each number, making it simpler to test for
**equality of numbers**

If the *exponent* of a number is negative, what do you need to do to the *mantissa* when converting from binary to decimal?

Fill in with 0s if the leftmost bit is a 0, and fill in 1s if its a 1.