Discrete Mathematics Midterm 1

Predicate

"is greater than 3," for example, refers to a property that the subject of the statement can have

Existential Quantification

"there exists an element x in the domain such that P(x)"

Fallacy of Denying the Hypothesis

((p -> q) ^ NOT p) -> NOT q

Ordered n-tuples

(a1, a2, a3, a4,...,an) is the ordered collection that has a1 as its first element, a2 as its second element, and an as its nth element

Associative Laws

A UNION (B UNION C) = (A UNION B) UNION C), A INTERSECT (B INTERSECT C) = (A INTERSECT B) INTERSECT C

Idemptotent Laws

A UNION A = A, A INTERSECT A = A

Commutative Laws

A UNION B = B UNION A, A INTERSECT B = B INTERSECT A

Complement Laws

A UNION NOT A = U, A INTERSECT NOT A = NULL

Domination Laws

A UNION U = U, A INTERSECT NULL = A

Argument

a sequence of statements that end with a conclusion, a sequence of propositions

Geometric Progression

a sequence of the form a, ar, ar^2, ...,ar^n

Bit string

a sequence of zero or more bits, the length of this string is the number of bits in the string

Subset

a set A is one of these for B if and only if every element of A is also an element of B, we use the notation A ⊆ B

Union of Collection

a set that contain those elements that are members of at least one set in the collection

Intersection of Collection

a set that contains those elements that remembers of all the sets in the collection

Countable

a set that is either finite or has the same cardinality as the set of positive integers, aleph is an infinite set that is countable

Null Set

a set with no elements, { }

Singleton Set

a set with one element

Corollary

a theorem that can be established directly from a theorem that has been proved

Proof

a valid argument that establishes the truth of a theorem

Quantification

a way to create a proposition from a propositional function

Congruence

a ≡ b (mod m)

Arithmetic Progression

a, a + d, a + 2d, ... , a + nd, where a is the initial term and d is the common difference

Closed Formula

an explicit formula for the terms of a sequence, rather than a recursive formula

Proof by Contraposition

an extremely useful type of indirect proof, make use of the fact that p -> q is equivalent to NOT q -> NOT p

Set

an unordered collection of objects, each called an element, said to contain its elements, A ∈ a indicates that A contains an element a

Showing f is Not Surjective

find a particular y ∈ B such that f(x) != y for all x ∈ A

Showing f is Not Injective

find particular elements x, y ∈ A such that x != y and f(x) = f(y)

Fibonacci Sequence

fn = fn-1 + fn-2

Function Equality

if two functions have the same domain, the same codomain, and map each element of their common domain to the same element in their common codomain

Z+

positive integers

R+

positive real numbers

Logic Circuit

receives input signals p1, p2,...,pn, each a bit, and produces output signals s1, s2,...,sn, each a bit

AND gate

takes two input signals p and q, each a bit, and produces as output the signal p ^ q

OR gate

takes two input signals p and q, each a bit, and produces as output the signal p ∨ q

Conclusion

the final proposition of an argument

Inverse Function

the function that assigns to an element b belonging to B the unique element a in A such that f(a) = b, denoted by f^-1

Existential Generalization

the rule of inference that is used to conclude that ∃ x P(x) is true when a particular element d with P(d) true is known

Universal Generalization

the rule of inference that states that ∀ x P(x) is true, given the premise that P(d) for all elements in the domain

Universal Instantiation

the rule of inference used to conclude that P© is true, where c is a particular member of the domain, given the premise ∀ x P(x)

Existential Instantiation

the rule that allows us to conclude that there is an element d in the domain for which P(d) is true if we know that ∃ x P(x) is true

Intersection

the set containing those elements in both A and B, A ∩ B

Power Set

the set of all subsets of the set S, P(S)

Axioms

the statements we assume to be true

Disjoint Sets

their intersection is the empty set

Backward Substitution

we began with an and iterated to express it in terms of falling terms of the sequence until we found it in terms of a1

Without Loss of Generality

when used in a proof, we assert that by proving one case of a theorem, no additional argument is required to prove other specified cases

DeMorgan's Laws

¬(p ^ q) ≡ ¬p ∨ ¬q, ¬(p ∨ q) ≡ ¬p ^ ¬q

Inverse

¬p → ¬q, equivalent to the converse

Negation

¬p, "it is not the case that p", read "not p"

Proper Subset

A is a subset of B but A != B, A ⊂ B

Difference of Sets

A - B, the set containing those elements that are in A but not in B, also called the complement of A with respect to B

Identity Laws

A INTERSECT U = A, A UNION NULL = A

Absorption Laws

A UNION (A INTERSECT B) = A, A INTERSECT (A UNION B) = A

Distributive Laws

A UNION (B INTERSECT C) = (A UNION B) INTERSECT (A UNION C), A INTERSECT (B UNION C) = (A INTERSECT B) UNION (A INTERSECT C)

Muddy Children Puzzle

A father says "at least one of you has a muddy forehead", then asks them "Do you know whether you have a muddy forehead?", after both children answer the first time, they realize they each have muddy foreheads

Domain/Codomain

If f is a function from A to B, we say that A is the domain of f and B is the codomain of f

Geometric Series Summation

Summation from j =0 to n of ar^j = (ar^(n+1) - a) / (r-1)

Conjecture

a statement that is being proposed to be a true statement, usually on the basis of some partial evidence

Conditional Statement/Implication

p → q, p implies q, "if p, then q," p is called the hypothesis, q is called the conclusion

Biconditional Statement

p ↔ q, "p if and only if q," true when p and q have the same truth values, and false otherwise, also called bi-implications

Disjunction

p ∨ q, "p or q", false when p and q are false and true otherwise

Propositional Function

p(x), the value of the function at x, can either be true or false

Modus Ponens/Law of Detachment

p, p->q, therefore q

Conjunction

p, q, p AND q

Addition

p, therefore p OR q

Propositional Logic

the area of logic that deals with propositions

Complement

the complement of A with respect to U, the universal set, represented by all elements outside the circle in a Venn Diagram, Ā

Valid

the conclusion, or final statement of the argument, must follow from the truth of the preceding statements of the argument

Union

the set that contains those elements that are either in A or in B, or in both, A ∪ B

Cardinality

the size of a set, i.e. the number of elements, |S|

Least Common Multiple

the smallest positive integer that is divisible by both a and b, lcm(a,b)

Venn Diagrams

used to represent sets graphically, the rectangle represents the universal set U

Dividend

q = a div d

Converse

q → p, equivalent to the inverse

Remainder

r = a mod d

Q

rational numbers

R

real numbers

Set Equality

two sets are equal if and only if they have the same elements

Composition of Functions

(f o g) = f(g(a)), let g be a function from the set A to the set B and let f be a function from the set B to the set C

Associative Laws

(p OR q) OR q ≡ p OR (q OR r), (p AND q) AND r ≡ p AND (q AND r)

Cartesian Product

A x B, the set of all ordered pairs (a,b) where a ∈ A and b ∈ B, can do this with more than 2 sets

Sequence

a function from a subset of the set of integers, use the notation an to denote the image of the inter n, an is a term

Invertible

a function that can have an inverse defined, a one-to-one

Injective

a function that is one-to-one

Surjective

a function that is onto

Lemma

a less important theorem that is helpful in the proof of other results

DeMorgan's Laws

Complement of (A INTERSECT B) = Complement of A UNION Complement of B

Complementation Law

Complement of complement of A is A

Rules of Inference

a set of some relatively simple argument forms

Division Algorithm

Let a be an integer and d a positive integer, then there are unique integers q and r, with 0 <= r < d such that a = dq + r

Inverter

NOT gate, takes an input bit p, produces as output ¬p

p -> q

NOT p OR q, NOT q -> NOT p

Modus Tollens

NOT q, p -> q, therefore NOT p

Double Negation Law

NOT(NOT p) ≡ p

Set Builder notation

O = {x ∈ Z | x is odd and x < 10} example, specify the set and then add conditions the elements must match

Theorem

a statement that can be shown to be true

Tautology

a compound proposition that is always true, no matter what the truth values of the propositional variables that occur in it

Contingency

a compound proposition that is neither a tautology nor a contradiction

Proposition

a declarative sentence that is either true or false, but not both (a sentence that declares a fact), questions, equations, and commands are not propositions

Circular Reasoning

a fallacy arises when a statement is proved using itself, or a statement equivalent to it

Onto/Surjection

a function f from A to B, if and only if for every element b ∈ B, there is an element a ∈ A with f(a) = b

One-to-One/Injunction

a function f is one of these if and only if f(a) = f(b) implies that a = b for all a and b in the domain of f

Partial Function

a partial function f from a set A to a set B is an assignment to each element a in a subset of A, called the domain of definition of f, of a unique element b in B

Trivial Proofs

a proof of p -> q that uses the fact that q is true

Bit

a symbol with two possible values, namely 0 and 1

Boolean variable

a variable whose value is either true or false

Recurrence Relation

an equation that express an in terms of one or more of the previous terms of the sequence, a solution satisfies its terms, recursively defines a sequence

Fallacies

incorrect reasoning that leads to invalid arguments

Z

integers

Proof by Cases

element-wise proof, proving each individual pi for i = 1,2,3,4,... individually, establish a number of possible cases and prove it for each

Universal Quantifier

asserts that P(x) is true for all values of x in this domain, the domain specifies the possible values of the variable x, "P(x) for all values of x in the domain"

Floor Function

assigns to the real number x the largest integer that is less than or equal to x, |_x_|

Ceiling Function

assigns to the real number x the smallest integer that is greater than or equal to x, [x]

Proofs by Contradiction

because (r ^ NOT r), we can proof NOT p -> (r ^ NOT r) by this kind of proof

Nested Quantifiers

everything inside the scope of a quantifier can be thought of as a propositional function, ∀ x ∃ y(x + y = 0) is equivalent to ∀ x Q(x) where Q(x) = ∃ y(x + y = 0)

Function

f from A to B is an assignment of exactly one element of B to each element of A, f(a) = b if b is the unique element of B assigned by the function, f: A -> B

Grand Hotel

bump each guest up a room, the first room is for the new guest

The Order of Quantifiers

can affect the truth value of a statement, for example "there is a real number y such that for every real number x, Q(x,y)" is false, but if the quantifiers were flipped it would be truwe

C

complex numbers

Contradiction

compound proposition that is always false

Showing f is Surjective

consider an arbitrary element y ∈ B and find an element x ∈ A such that f(x) = y

Truth Tables of Compound Propositions

construct n^2 rows, for n propositions, make columns for each part of the compound proposition you need

Bit operations

correspond to logical connective, AND, OR, XOR

Roster Method

describing a set using all its elements, for example the set of the vowels V = {a,e,i,o,u}

Common Mistakes in Proofs

division by zero, invalid rules of inference,

Image/Preimage

f(a) = b, b is the image, a is the preimage

Relatively Prime

gcd = 1

Precedence of Logical Operators

generally we use parentheses to specify this, but we can use an ordering to reduce the number of parentheses, apply negation first, then and/or, then implies

Truth Table

has a row for each of the two possible truth values of a proposition p, each row rows the truth value of ¬p for the corresponding value

Vacuous Proof

if we can show NOT p, then we have a proof of this kind of p -> q

Union of Countable

if A and B are countable sets, then A U B is also countable

Schroder-Bernstein Theorem

if A and B are sets with |A| <= |B|, and |B| <= |A|, then |A| = |B|

One-to-One Correspondence/Bijection

if a function is both one-to-one and onto

Graph

if f is a function from the set A to the set B, the set of ordered pairs {(a,b) | a ∈ A and f(a) = b}

Logically equivalent

if p ↔ q is a tautology, p ≡ q

Satisfiable

if there is an assignment of truth values to its variables that makes it true

Direct Proof

p -> q, is constructed when the first step is the assumption that p is true, subsequent steps are constructed using rules of inference, with the final step showing that q must also be true

Hypothetical Syllogism

p -> q, q -> r, p -> r

Identity Laws

p AND T ≡ p, p OR F ≡ p

Simplification

p AND q, therefore p

Absorption Laws

p OR (p AND q) ≡ p, p AND (p OR q) ≡ p

Domination Laws

p OR T ≡ T, p AND F ≡ F

Idempotent Laws

p OR p ≡ p, p AND p ≡ p

Commutative Laws

p OR q ≡ q OR p, p AND q ≡ q AND p

Resolution

p OR q, NOT p OR r, therefore q OR r

Disjunctive Syllogism

p OR q, NOT p, Q

Distributive Laws

p Or (q AND r) ≡ (p OR q) AND (p OR r), p AND (q OR r) ≡ (p AND q) OR (p AND r)

Greatest Common Divisor

largest integer d such that d | a and d | b, denoted by gcd(a,b)

Propositions

less important theorems

Propositional Variables

letters that represent propositions, just as letters are used to denote numeral variables, generally we use p,q,r,s...

Conjunction

p ^ q, "p and q", true when both p and q are true, false otherwise

Exclusive or

p or q but not both, p ⊕ q

Inclusive or

p or q, true when at least one of the true positions is true

Factorial Function

n! = (n) * (n-1) * (n-2) * ... * (1)

N

natural numbers

Compound Propositions

new propositions formed from editing prositions using logical operations

Begging the Question

occurs when one or more steps of a proof are based on the truth of the statement being proved

Induction

proof a base case for n, then proof n+1

Indirect Proofs

proofs of theorems of the form ∀ x (P(x) -> Q(x)), proofs that do not start with the premises and end with the conclusion

Exhaustive Proof

proofs that can be prone by examining a relatively small number of examples, show for each number

Boolean searches

searches employing techniques from propositional logic, connective AND is used to match records that contain both of two search terms, etc.

Cardinality

sets have the same this if there is a one-to-one correspondence from A to B

Showing f is Injective

show that f(x) = f(y) for arbitrary x, y ∈ A with x != y, then x = y

Summation Notation

sigma, used to sum from the lower limit to the upper limit all of the terms in the sequence, j is called the index of summation

Premises

statements of an argument

Forward Substitution

successive terms beginning with the initial condition and ending with an

Fallacy of Affirming the Conclusion

treat arguments with premises p -> q and q and conclusion p as a valid argument form, which it is not

Truth Value

true, denoted by T, if it is a true proposition, and false, denoted by F, if it is a false proposition

Equivalent

two compound propositions are this if they always have the same truth value

Uniqueness Proof

two parts: we show that an element x with the desired property exists, we show that if y != x, then y does not have the desired property

A divides B

true if there is an integer c such that b = ac

Informal Proofs

where more than one rule of inference may be used in each step, where steps may be skipped, where the axioms being assumed and the rules of inference used are not explicitly stated

Contrapositive

¬q → ¬p, equivalent to the original statement

DeMorgan's Laws for Quantifiers

¬∀ x P(x) ≡ ∃ x ¬ P(x), ¬∃ x Q(x) ≡ ∀ x ¬ Q(x)

for all

∀

there exists

∃

Existence Proofs

∃ x P(x), finding a witness to prove it is constructive, deconstructive is to prove it without finding a witness

Uniqueness Quantifier

∃!, "there exists a unique x such that P(x) is true"