# Discrete Mathematics

## 40 terms

### Set

A well-defined collection of objects.

### Subset

Let A be a set. A set B is called a _______ of A, denoted by B c A, provided that for every x belonging to B, x also belongs to A.

### Union

Let A and B be two sets. The _______ of A and B, denoted by A u B = {x | x belongs to A or x belongs to B}

### Intersection

Let A and B be two sets. The _______ of A and B, denoted by A n B = {x | x belongs to A and x belongs to B}

### The Empty Set

A set with no elements is called _______ and is denoted by phi. _______ is a subset of every set.

### Equality

Two sets A and B are said to have _______, denoted by A=B, provided A c B and B c A.

### Cartesian Product

Let A be a non-empty set. The _______ of A with itself is denoted by A x A = {a1, a2 | a1 and a2 both belong to A}

### Relation

Let A be any non-empty set. A _______ on A is a subset of A x A.

### Power Set

The _______ of A, denoted by P(A), is given by P(A) = {x | x is subset or equal to A} and represents the collection of all possible subsets of a set.

### Cardinality

Let A be a finite set. The _______ of A is the number of elements in A. It is denoted by |A|.

### Disjoint

Let U be a non-empty set. Two subsets A and B are said to be _______ provided A n B = the empty set.

### Partition

Let U be a non-empty set. A collection of subsets (A1, A2, A3..., An) is said to be a _______ of U provided that
1) U to the Ai, with i going from zero to n, is equal to U.
2) Ai n Aj = the empty set AND 1<= i, j<= n, and i does not equal j.

### Reflexive

A relation R on A is said to be _______ provided for every x belonging to A, (x, x) belongs to R.

### Symmetric

A relation R on A is said to be _______ provided for any x, y belonging to A, if (x, y) belongs to R, then (y, x) also belongs to R.

### Transitive

A relation R on A is said to be _______ provided for every x, y, z belonging to A, if (x, y) belongs to R and (y, z) belongs to R, then (x, z) also belongs to R.

### Equivalence Relation

A relation on a set A is called an ________ provided R is reflexive, transitive, and symmetric.

### Difference

Let A and B be two sets. The _______ between these two sets is denoted by A - B = {x | x belongs in A, x does not belong in B}

### Equivalence Class

Let A be a non-empty set and R be an equivalence relation on A. Let x be any arbitrary element of A. Then the _______ of x, denoted by [x] = {a belongs to A | (x, a) belongs to R}

### Division Algorithm

Let m and n be two integers (n cannot equal zero). Then there exist integers q and r such that m = nq + r, 0 <= r < n.
m is the dividend
n is the divisor
q is the quotient
r is the remainder

### Function

Let A and B be two non-empty sets. A _______ from A to B, denoted by f: A -> B, is a rule that assigns to each a belonging to A a unique b belonging to B, called f(a).

### Range

Let f: A -> B be a function. Then the _______ of f is given by the ran(f) = {f(x) | x belongs to A} is subset or equal to B.

### One-To-One (1-1)

Let f: A -> B be a function. f is said to be _______ provided for x1, x2 belonging to A, if f(x1) = f(x2), then x1 = x2. Equivalently, if x1 does not equal x2, then f(x1) does not equal f(x2).

### Onto

Let f: A -> B be a function. f is said to be _______ provided for every b belonging to B, there exists and a in A such that f(a) = b. For each element in the co-domain, there exists an element in the domain that is mapped to the given co-domain element).

### Bijection/1-1 Correspondence

A function f: A -> B is said to set a _______ provided f is 1-1 and onto.

### Finite

A set X is said to be finite provided there exists an n in N such that the function f: X -> {1, 2, 3..., n} is a bijection.

### Equality of Cardinality

Let A and B be any two sets. If f: A -> B is a bijection, then |A| = |B|.

### Infinite

A set X is said to be _______ provided it is in 1-1 correspondence with its proper subset.

### Countably Infinite

A set A is said to be _______ provided A is in bijective correspondence with |N.
f: A -> N, f is 1-1 and onto

### Shoder-Bernstein Theorem

Let A and B be two sets. If there exists 1-1 functions f: A -> B and g: B -> A, the there exists a bijective function from A to B.

### Factorial

Let n be a non-negative integer. Then the _______ of n, denoted by n!, is equal to 1 if n = 0 or 1, and n(n-1)(n-2)(n-3)....[n-(n-1)].

### Permutation

Let X be a finite set. A _______ of X is a bijective function f: X->X (1-1 correspondence).

### Prime

A natural number p is said to be _______ provided the only divisors of p are 1 and p.

### Composite

A natural number is said to be _______ if it is not a prime.

### GCD

Let a and b belong to set Z, both values unequal to zero. An integer g is called the _______ of a and b provided g|a and g|b and if c is any other divisor of a and b, c<=g.

### Co-prime

Two integers a and b are said to be _______ provided gcd(a,b) = 1.

### LCM

Let a and b be two integers. The _______ of a and b is a common multiple of a and b such that if h is any other common multiple of an and b, the _______ is less than or equal to h.

### Euler's Phi function

For n >= 1, the _______ gives the number of positive integers less than n that are relatively prime to n.

### Euler's theorem

Let n >= 1 and a be an integer such that gcd(a,n) = 1. Then a^(phi(n)) is congruent to 1(modn).

### Wilson's theorem

If p is any prime, then p-1! is congruent to -1(modp).

### Fermat's Little theorem

If p is a prime such that p does not divide a, then a^(phi(p)) is congruent to 1(modp). Observe that for n > 1, p(n) = n-1 if and only if n is a prime.