IB HL Discrete Mathematics

The degree sequence is...

(of a graph) the list of degrees of vertices of a graph, including duplicates.

Two numbers are relatively prime if...

...their gcd = 1

A complete graph K_n has how many edges?

1/2 x n(n-1)

An Eulerian circuit is...

An Eulerian circuit is a circuit which contains no repeated edges. (i.e. goes back to where it starts)

An Eulerian trail is...

An Eulerian trail is a trail which contains no repeated edges.

What does the Nearest Neighbour algorithm give us?

An upper bound for the Travelling Salesman problem.

If there are kn + 1 elements divided into n categories, what can be deduced and why?

By the Pigeonhole Principle, at least one category must contain at least k + 1 elements.

If there are n + 1 elements divided into n categories, what can be deduced and why?

By the Pigeonhole Principle, at least one category must contain at least two elements.

If x = a (mod m) and x = a (mod n) then

If x = a (mod m) and x = a (mod n) then x = a (mod mn)

What is K_r,s and how many edges does it have?

K_r,s is the complete bipartite graph with r vertices in one set and s in the other. It has rs edges.

How do you find a minimum spanning tree?

Kruskal's algorithm.

How do we find the closed form of the recurrence relation u(n+1) = p un + q?

Let un = c x p^n + d. Then u(n+1) can be expressed in the same form and combined with the recurrence form.

State the four rules that follow if a ≡ b and c ≡ d (mod m)

Multiplication: ka ≡ kb (mod m) Addition: a + c ≡ b + d (mod m) Cross Multiplication: ac ≡ bd (mod m) Power: a^n ≡ b^n (mod m)

What makes a simple graph?

No multiple edges, no loops.

What proof method would you use to show that the square root of 2 is irrational?

Proof by contradiction.

What are the steps of Kruskal's algorithm?

Remember to list the order in which edges are added.

What are the steps for the Nearest Neighbour algorithm?

Remember to list the order in which the edges are added.

If 30 elements are divided between four categories, what can be deduced?

Since 30 = 4 x 7 + 2, at least one category must contain at least 8 elements. (We have one more element that we need to prove this.)

If proving a strong induction statement related to the recurrence relation u_n=u_(n-2)+u_(n-5), how many initial statements would you need to show are correct?

Six

What is the Chinese Postman problem?

The Chinese Postman has to find the shortest route around a graph visiting every edge at least once and returning to the start.

How do we know that any number N > 1 can be expressed as a unique product of prime numbers?

The Fundamental Theorem of Arithmetic states that any number N > 1 can be expressed as a unique product of prime numbers.

The Fundamental Theorem of Arithmetic states that

The Fundamental Theorem of Arithmetic states that any number N > 1 can be expressed as a unique product of prime numbers.

What do we know about the number of odd vertices in any graph, and why?

The Handshaking Lemma states that the number of odd vertices in a graph must be even.

What is the Travelling Salesman Problem?

The Travelling Salesman has to visit every vertex and return to the start by the shortest possible route.

How do you test for divisibility by 11?

The alternating sum of its digits is divisible by 11. e.g. 3942985: 3 - 9 + 4 - 2 + 9 - 8 + 5 = 2 which is not divisible by 11.

The degree of a vertex is...

The degree of a vertex is the number of edges it has. A loop counts twice.

If 120 elements are divided between 10 categories, what can be deduced?

120 = 11 x 10 + 10, so at least one category must contain at least 12 elements. (We have nine more elements that we need, but this is not enough to prove that at least one category must contain at least 13 elements, which is quite obviously not the case).

A Hamiltonian cycle...

A Hamiltonian cycle visits every vertex exactly once, then returns to the starting vertex.

A Hamiltonian Path...

A Hamiltonian path visits every vertex exactly once.

A bipartite graph is...

A bipartite graph is a graph such that the vertices can be divided into two sets where no edge joins two vertices in the same set.

A circuit is...

A circuit is a trail which starts and ends at the same vertex.

A complete graph is...

A complete graph is a graph where an edge exists between every pair of distinct vertices. Notation is K_n. for a complete graph with n vertices.

A cycle is...

A cycle is a path which starts and ends at the same vertex.

A subgraph of G is...

A graph all of whose vertices and edges are in G.

A graph has an Eulerian circuit if and only if...

A graph has an Eulerian circuit if and only if it has no odd vertices (and is connected.) aka Eulerian.

A graph has an Eulerian trail if and only if...

A graph has an Eulerian trail if and only if it has exactly two odd vertices (and is connected.) aka Semi-Eulerian

A graph is planar if...

A graph is planar if it can be drawn without any crossing edges.

What does the Deleted Vertex algorithm give us?

A lower bound for the Travelling Salesman problem.

A path is...

A path is a walk with no repeated vertices.

A trail is...

A trail is a walk with no repeated edges.

A tree is...

A tree is a graph which contains no cycles.

A walk is...

A walk is any sequences of adjacent edges

If gcd(a,b)=d then gcd(a,a - b) = ?

If gcd(a,b)=d then gcd(a,a-b) = d

How do you find the shortest path between two vertices?

Dijkstra's algorithm.

Euclid's Lemma states that...?

Euclid's Lemma states that if a prime p | ab then either p | a or p | b.

How do we know that if prime p | ab then either p | a or p | b?

Euclid's Lemma states that if a prime p | ab then either p | a or p | b.

What does Kruskal's algorithm do?

Find a minimum spanning tree.

What does Dijkstra's algorithm do?

Find the shortest path between two vertices.

When is a^p ≡ a (mod p) true?

For all integers a and primes p.

Fermats Little Theorem states that

For integer a and prime p: a^p ≡ a (mod p) a^(p-1) ≡ 1 (mod p) if and only of a is not a multiple of p

When is a^(p-1) ≡ 1 (mod p) true?

For integers a and primes p where a is not a multiple of p.

How are the number of edges in G and G' related?

G' is the complement of G. The number of edges in G and G' sum to the number of edges in K_n where n is the number of vertices in G and G'.

If a number in base n is divisible by n - 1 then...

If a number in base n is divisible by n - 1 then the sum of its digits is also divisible by n - 1. The converse is also true, and you may be asked to prove this.

If a | N and b | N, what else divides N?

If a | N and b | N, then lcm(a,b) | N

If a ≡ b (mod m) and d | a and b, and gcd(d,m) = 1 then

If a ≡ b (mod m) and d | a and b and gcd(d,m) = 1 then a/d ≡ b/d (mod m)

State the division rules for congruences, i.e. when are you allowed to divide a congruence a ≡ b (mod m)

If a ≡ b (mod m) and d | a and b and gcd(d,m) = 1 then a/d ≡ b/d (mod m) If a ≡ b (mod m) and d | a and b and m, and gcd(d,m) = 1 then a/d ≡ b/d (mod m/d) If a ≡ b (mod m) and d | a and b and m, and gcd(d,m) ≠ 1 then a/d ≡ b/d (mod m/(gcd(d,m)) In general, if you can divide a and b by an integer d, you can do so, but you must divide m by gcd(d,m) if you can.

If a ≡ b (mod m) and d | a and b and m, and gcd(d,m) = 1 then

If a ≡ b (mod m) and d | a and b and m, and gcd(d,m) = 1 then a/d ≡ b/d (mod m/d)

If a ≡ b (mod m) and d | a and b and m, and gcd(d,m) ≠ 1 then

If a ≡ b (mod m) and d | a and b and m, and gcd(d,m) ≠ 1 then a/d ≡ b/d (mod m/(gcd(d,m))

If a ≡ b and c ≡ d (mod m) then the addition rule states that:

If a ≡ b and c ≡ d (mod m) then the addition rule states that a + c ≡ b + d (mod m)

If a ≡ b and c ≡ d (mod m) then the cross multiplication rule states that:

If a ≡ b and c ≡ d (mod m) then the cross multiplication rule states that ac ≡ bd (mod m)

If a ≡ b and c ≡ d (mod m) then the multiplication rule states that:

If a ≡ b and c ≡ d (mod m) then the multiplication rule states that ka ≡ kb (mod m)

If a ≡ b and c ≡ d (mod m) then the power rule states that:

If a ≡ b and c ≡ d (mod m) then the power rule states that a^n ≡ b^n (mod m) for all natural numbers N.

If gcd(a,b)=d then gcd(a,a - qb) = ?

If gcd(a,b)=d then gcd(a,a - qb) = d

What is the first step for finding a closed form of the recurrence relation u(n+2) = a x u (n+1) + b x u(n)?

The first step for finding a closed form of the recurrence relation u(n+2) = a x u (n+1) + b x u(n) is solving the auxilliary equation k^2 - ak - b = 0.

The complement of G is...

The graph G' which has the same set of vertices as G, but edges that exist in G do not exist in G', and vice versa.

How do you test for divisibility by 8?

The number formed by the last three digits is divisible by 8.

How do you test for divisiblity by 4?

The number formed by the last two digits is divisible by 4.

The Handshaking Lemma states that...

The sum of degrees of the vertices of a graph is twice the number of edges.

What does it mean to say vertices a and b are connected?

There is a path connecting a and b.

What does it mean to say vertices a and b are adjacent?

There is an edge directly connecting a and b.

How does strong induction differ from weak induction?

Weak induction proves that a statement is true for n = k + 1 if the statement is true for n = k. Strong induction proves that a statement is true for n = k + 1 if the statement is true for n ≤ k. For strong induction you may have to prove more than just n = 1 for the first step.

When finding the closed form for u(n+2) = a x u (n+1) + b x u(n), if the auxilliary equation gives a repeated solution k, how do you proceed?

When finding the closed form for u(n+2) = a x u (n+1) + b x u(n), if the auxilliary equation gives a repeated solution k, use the form u_n = c1 x k^n + c2 x n x k^n.

When finding the closed form for u(n+2) = a x u (n+1) + b x u(n), if the auxilliary equation gives complex solutions, how do you proceed?

When finding the closed form for u(n+2) = a x u (n+1) + b x u(n), if the auxilliary equation gives complex solutions, the closed form is given by un = r^n (c1 cos (nθ) + c2 sin(nθ))

If ax + by = c has solutions x = p and y = q, what is the general solution and what condition exists to use this technique?

You must divide the original equation by gcd(a,b). Assuming this has been done, the general solution is given by x = p + kb and y = q - ka. It is easy to slip up here so check that your solution works for 1-2 values of k.

How do we know if ax + by = c has integer solutions for x and y?

ax + by = c has integer solutions for x and y if and only if gcd(a,b) | c.

gcd(a,b) x lcm(a,b) = ?

gcd(a,b) x lcm(a,b) = ab

If gcd(a,b)=d then lcm(a,b) = ?

lcm(a,b) = ab / gcd(a,b)