Discrete Math 189
Set intersection operation
The intersection of A and B, denoted A ∩ B, is the set of all elements that are elements of both A and B. A = { a, b, c, e, f } B = { d, e, f, g } A ∩ B = {e, f}
power set
The power set of a set A, denoted P(A), is the set of ALL subsets of A. For example, if A = { 1, 2, 3 }, then: P(A) = { ∅, { 1 }, { 2 }, { 3 }, { 1, 2 }, { 1, 3 }, { 2, 3 }, { 1, 2, 3 } } - ∅ has a size of 0 - { 1 }, { 2 }, { 3 } have sizes of 1 - { 1, 2 }, { 1, 3 }, { 2, 3 } have sizes of 2 and so on...
{0,1}^n
The set of binary strings of length n
alphabet
The set of characters used in a set of strings example: A = {x, y} - the alphabet is {x, y} used in this set of strings: A^2 = { xx, xy, yx, yy }.
concatenation of strings
joining the values of strings together example: If s = 010 and t = 11, then st = 01011 Concatenating any string x with the empty string gives back x: xλ = x.
negation of greater than or equal to
less than (negation of greater than is less than or equal to)
divide formula
m/n = k m is a multiple of n if there exists an integer k such that m = kn (or m/n = k is a whole number). n divides m if m/n = k is a whole number.
proposition
mathematical statement which is either always true or always false
f: X → Y
notation to express the fact that f is a function from X to Y. Set X is called the domain of f, and the set Y is the target (or co-domain) of f. The fact that f maps x to y (or (x, y) ∈ f) can also be denoted as f(x) = y.
ordered triple
ordered list of three items
proof by contrapositive
proves a conditional theorem of the form p → c by showing that the contrapositive ¬c → ¬p is true. In other words, ¬c is assumed to be true and ¬p is proven as a result of ¬c.
finite set
set that is either empty or whose elements can be numbered 1 through n for some positive integer n.
length of a string
the number of characters in a string
Cardinality
the number of elements in a set |A| Ex: If A = { 2, 4, 6, 10 }, then |A| = 4. The cardinality of the empty set |∅| is zero.
domain of a variable in a predicate
the set of all possible values for the variable
For function f: X → Y, an element y is in the range of f if and only if
there is an x ∈ X such that (x, y) ∈ f. Expressed in set notation: Range of f = { y: (x, y) ∈ f, for some x ∈ X }
Two sets are equal if
they have exactly the same elements
∀
universal quantifier, for all
superscript +
used to indicate the positive elements of a particular set (e.g. the set R+ is the set of all positive real numbers, and Z+ is the set of all positive integers) VICE VERSA with negative numbers 0 ∉ Z+ and 0 ∉ Z-
without loss of generality (WLOG or w.l.o.g.)
used to narrow the scope of a proof to one special case in situations when the proof can be easily adapted to apply to the general case Example: Theorem: For any two integers x and y, if x is even or y is even, then xy is even. Proof: Without loss of generality, assume that x is even. Then x = 2k for some integer k. Plugging in the expression 2k for x in xy gives xy = 2ky = 2(ky). Since k and y are integers, ky is also an integer. Since xy is equal to two times an integer, xy is even
If A = { 2, 3, 5, 7, 14 } is |P(A)| = 32 correct?
|A| = 5, so |P(A)| = 2^(5).
existential quantifier
∃ "there exists"
∀x P(x) ∧ Q(x) is equivalent to
(∀x P(x)) ∧ Q(x) And NOT ∀x (P(x) ∧ Q(x)). The quantifiers ∀ and ∃ are applied before the logical operations (∧, ∨, →, and ↔) used for propositions.
Capital vs lower case letters (sets)
- capital letters will be used as variables denoting sets - lower case letters are used for elements in the set - Variables can also represent an element of a set
Domain vs co-domain vs range
- domain: what can go into a function. - codomain: what may possibly come out of a function. - range: what actually comes out of a function.
t = 001. What is t1?
0011 (bits like 1 and 0 can be their own symbols)
subsets vs not subsets, vs proper subsets (example)
1. B = {2, 4} is a subset of A = {1, 2, 3, 4}. In a Venn diagram, the oval for B is inside the circle for A. Since 3 ∈ A and 3 ∉ B, B is a proper subset of A (B ⊂ A). 2. C = {2, 4, 5} is not a subset of A (C ⊈ A) because 5 ∈ C and 5 ∉ A. In the Venn diagram, the circle for C is not contained in the circle for A. 3. If D = {1, 2, 3, 4}, D ⊆ A and A ⊆ D, which implies that A = D. D is not a proper subset of A (D ⊄ A).
Statement: x + 1 < x
1. The proof starts with the inequality x + 1 < x and subtracts x from both sides to get 1 < 0. 2. Since 1 < 0 is false, the inequality x + 1 < x is false for every x and therefore ∃x(x + 1 < x) is false.
biconditional statement
a statement that can be written in the form "p if and only if q"
proposition formula (expression)
a type of syntactic formula which is well formed and has a truth value.
roster notation
a way of representing a set by listing the elements between braces
counterexample
an element in the domain for which the predicate is false
quantifiers
an expression (e.g., all, some ) that indicates the scope of a term to which it is attached, e.g. (for all, for every, for each, for any) or (for some, there is, there exists, there is at least one).
A ⊆ A x B True or False
False: The elements in A are single numbers such as 1. The set A x B contains only ordered pairs such as (1, x). Therefore A and A x B have no elements in common.
p ⇒ q
IF p is true THEN q is true
Divides definition
an integer n divides the integer m if: m/n = k is also an integer; division with ONLY integers example: 3 divides 15 because 5 (5 is results from 3/15) is also an integer
p ⇔ q
both p ⇒ q AND q ⇒ p simultaneously.
Two sets are equal if and only if
each is a subset of the other: A = B if and only if A ⊆ B and B ⊆ A
λ
empty string (string length = 0) {0, 1}^0 = {λ}.
contradiction
if the proposition is always false; opposite of tautology
∈
is an element in the set of example in 2 ∈ A means 2 is in the set of A. The symbol ∉ indicates that an element is not in a set.
f is not well-defined if
it maps an element of the domain (X) to zero elements or more than one element of the target
When to use a direct proof vs. a proof by contrapositive
Deciding involves some trial and error: Statement example: For every integer x, if x2 is even, then x is even. A direct proof assumes that x2 is even, which in mathematical terms means that x2 = 2k, for some integer k. Deriving an expression for x requires taking the square root of both sides, and it is not clear how to reason that sqrt(2k) is an even integer. Alternatively, a proof by contrapositive assumes that x is odd, which in mathematical terms means that x = 2k + 1, for some integer k. The expression for x can then be plugged into x2 resulting in an expression that is much easier to reason about.
Set of sets
A = { { 1, 2 }, ∅, { 1, 2, 3 }, { 1 } } Set A has 4 elements ONLY; 1 ∉ A, but { 1 } ∈ A. It is a set with other sets inside. A set can contain a combination of numbers and sets of numbers as in: B = { 2 , ∅, { 1, 2, 3 }, { 1 } } 2 ∈ B, so { 2 } ⊆ B. 1 ∉ B so { 1 } ⊈ B.
Proposition/Statement
A complete sentence or statement that can be evaluated as true or false.
Venn Diagram
A diagram that uses circles to display elements of different sets. Overlapping circles show common elements.
function
A function f that maps elements of a set X to elements of a set Y, is a subset of X × Y such that for every x ∈ X, there is exactly one y ∈ Y for which (x, y) ∈ f.
predicate
A logical statement whose truth value is a function of one or more variables (if a statement contains a variable, the statement is still considered to be a predicate and not a proposition)
arrow diagram
A pictorial representation of a conditional statement.
universal set venn diagram
A rectangle is used to denote the universal set U, and oval shapes within the rectangle are used to denote sets within U.
string
A sequence of elements/characters in a n-tuple set written without the usual punctuation (parentheses and commas). example: A = {x, y} A^2 would be { xx, xy, yx, yy }.
co-domain
A set from which the function's output values are chosen.
De Morgan's law
A set of rules for converting an expression containing NOTs into an expression that does not contain any NOTs. (ALMOST everything can be flipped so that it equals the original statement)
binary string and bit
A string whose alphabet is {0, 1}. bit is a character in a binary string. A string of length n is also called an n-bit string
proper subset
A subset that does not contain every element in another set; denoted by ⊂ and not ⊆.
truth table
A table used as a convenient method for organizing the truth values of statements
free variable
A variable x in the predicate P(x) variable is free to take on any value in the domain
Number Systems
A way of counting things, and identifying quantity. The set of symbols used to express quantity as the basis for counting.
A = { x^2 | x∈R }
All positive real values (all the values of y in this graph)
symbolic logic
An application of mathematical methods to the processes of thought; uses conventional symbols to represent terms, propositions & relations among them
ordered n-tuple
An ordered list of 4 or more items example: (w, x, y, z) is an ordered 4-tuple and (u, w, x, y, z) is an ordered 5-tuple
element chasing proof (with subset theorems)
Assume that x is in the set of the left side of the theorem and then prove that x is in the right side of the theorem. Prove A ⊆ B: Assume x ∈ A to be true Show that x ∈ B.
∀x vs ∃x
Every value of the domain works vs. There exists a value(s) in the domain that works.
A x B x C ⊆ A x B x C x D True of False
False: The set A x B x C contains triples. The set A x B x C x D contains 4-tuples. The two sets do not have any elements in common.
subset
If every element in A is also an element of B, then A is a subset of B, denoted as A ⊆ B. If there is an element of A that is not an element of B, then A is not a subset of B, denoted as A ⊈ B. If the universal set is U, then for every set A: ∅ ⊆ A ⊆ U
Cartesian product of a set A^n
If set A = {0, 1}, then A^n is the set of all ordered n-tuples whose entries are bits (0 or 1). If n = 3: {0, 1}^3 = { (0, 0, 0), (0, 0, 1), (0, 1, 0), (0, 1, 1), (1, 0, 0), (1, 0, 1), (1, 1, 0), (1, 1, 1) }
Is the following statement true? For any two sets, X and Y, if X ⊆ Y, then X ⊂ Y.
NO; It is possible that X = Y in which case X ⊆ Y, but X ⊄ Y.
If A = { 2, 3, 5, 7, 14 } is 2 ∈ P(A) correct?
NO; The elements of P(A) are all subsets of A. 2 is an element of A, which is written as 2 ∈ A. P(A) includes { 2 }, so 2 ∉ P(A), BUT { 2 } ∈ P(A).
Empty Set (∅, {}, null set)
a set with no elements The empty set is sometimes referred to as the null set and can also be denoted by {}. Because the empty set has no elements, for any element a, a ∉ ∅ is true.
how to use contrapositive proofs with multiple hypothesis
Prove that one of the hypotheses is false. The first step is to assume that the rest of the hypotheses are true and the conclusion is false
OR - Disjunction
Represented with symbol ∨.
NOT - Negation
Represented with the symbol: ¬ or ~. This negates a proposition
AND - Conjunction
Represented with the symbol: ∧ or ⋅.
Logical Equivalence
Sentences p and q are logically equivalent if and only if it is not possible for one of these sentences to be true while the other sentence is false.
T or F: A statement with no free variables is a proposition
T because the statement's truth value can be determined
set complement
The complement of a set A (denoted with a line like this: Ā) is the set of all elements in U that are not elements of A. An alternative definition of Ā is U - A. For example, let U = Z, and define: A = { x ∈ Z: x is odd }
set difference
The difference between two sets A and B, denoted A - B, is the set of elements that are in A but not in B. A = { a, b, c, e, f } B = { d, e, f, g } A - B = { a, b, c }
empty set ∅ vs { ∅ }
The empty set ∅ is not the same as { ∅ }. The cardinality of { ∅ } is one since it contains exactly one element, which is the empty set.
symmetric difference
The symmetric difference between two sets, A and B, denoted A ⊕ B, is the set of elements that are a member of exactly one of A and B, but not both. A ⊕ B = ( A - B ) ∪ ( B - A )
Set union operation
The union of two sets, A and B, denoted A ∪ B , is the set of all elements that are elements of A or B. A = { a, b, c, e, f } B = { d, e, f, g } A ∪ B = {a, b, c, e, f, d, g}
bound variable
The variable x in the statement ∀x P(x) variable is bound to a quantifier.
if set A = { a, b, c }, is this True or False? (a, b, b, a) ∈ A^4
True: (a, b, b, a) is a 4-tuple whose entries are all elements of A.
Sets with many (or infinite) elements
Use ellipses (...) to denote it B = { 1, 3, 5, ... , 99 } or C = { 3, 6, 9, 12, .... }
Proof by Contradiction (Indirect Proof)
Where you prove a theorem true by showing that the opposite is not true. It starts by assuming that the theorem is false. Then show that the proof of this assumption has inconstancies; if the assumption that the theorem is false leads to a conclusion which cannot be true, then the theorem must be true.
Is the following statement true? For any two sets, X and Y, if X ⊂ Y, then X ⊆ Y.
YES; If X ⊂ Y, then every element of X is also an element of Y, so X ⊆ Y.
set
a collection of objects called elements The order in which the elements are listed is unimportant. Repeating elements are also unimportant (you can just write one element to represent all the repeating ones)
tautology
a formula or assertion that is true in every possible interpretation
ordered pair
a pair of numbers written as (x,y) items placed in ( ) means that the order matters, unlike with { }. The first entry of the ordered pair (x, y) is x and the second entry is y.
proof by cases
a proof broken into separate cases, where these cases cover all possibilities
direct proof
a proof that starts with the given statements and uses the laws of logic to arrive at the statement to be proved
existentially quantified statement
a proposition that is true if and only if P(n) is true for at least one value n in the domain of variable x ∃x P(x)
Cartesian product
a set of pairs (x, y) of elements from two sets X and Y.
universal sets
a set that contains all elements mentioned in a particular context (usually denoted by the variable U) In a discussion about certain types of real numbers, the universal set would be all real numbers. In a discussion about the academic standing of certain students at a school, the universal set would be the set of all students enrolled in the school.
set builder notation
a set that is defined by specifying that the set includes all elements in a larger set that also satisfy certain conditions. A = { x ∈ S : P(x) } - S is the larger set, all its elements included in set A - P(x) is some condition for membership in A - The colon symbol ":" is read "such that". The description for A above would read: "all x in S such that P(x)".
infinite set
a set that is not finite
