CSCE222 First Exam

Existential Generalization (EG)

"Michelle got an A in the class." "Therefore, someone got an A in the class."

even integer

2k

Tuples

The ordered n-tuple (a1,a2,.....,an) is the ordered collection that has a1 as its first element and a2 as its second element and so on until an as its last element

pre-image

if the x for F(x)

Floor function

|_x_| is the largest integer less than or equal to x. 3.5=3

Implication (symbol)

→

Existential Instantiation (EI)

"There is someone who got an A in the course." "Let's call her a and say that a got an A"

De Morgan's Second Law

(A ∩ B)' = (A)' ∪ (B)'

De Morgan's First law

(A ∪ B)' = (A)' ∩ (B)'

Complementation law

(A=)=A *first equal sign denotes a double bar on top of A*

ordered pairs

2-tuples are called ordered pairs.

How many rows are there in a truth table with n propositional variables

2^(n)

odd integer

2k+1

if a set has n elements what is the cardinality of the *power set*

2ⁿ

Bijections

A function f is a one-to-one correspondence, or a bijection, if it is both one-to-one and onto (surjective and injective). *basically a function as we usually know*

Geometric Progression

A geometric progression is a sequence of the form: where the initial term a and the common ratio r are real numbers.

Partial Functions

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 ex: f: N → R where f(n) = √n is a partial function from Z to R where the domain of definition is the set of nonnegative integers. Note that f is undefined for negative integers.

Proper Subsets

A proper subset of a set A is a subset of A that is not equal to A. In other words, if B is a proper subset of A, then all elements of B are in A but A contains at least one element that is not in B. For example, if A={1,3,5} then B={1,5} is a proper subset of A. The set C={1,3,5} is a subset of A, but it is not a proper subset of A since C=A. The set D={1,4} is not even a subset of A, since 4 is not an element of A.

Sequences

A sequence is a function from a subset of the integers (usually either the set {0, 1, 2, 3, 4, .....} or {1, 2, 3, 4, ....} ) to a set S.

Complement laws

A ∪ A' = U A ∩ A' = ∅

Algorithms

An algorithm is a finite set of precise instructions for performing a computation or for solving a problem.

Proof by Contraposition

Assume ¬q and show ¬p is true also. This is sometimes called an indirect proof method. If we give a direct proof of ¬q → ¬p then we have a proof of p → q.

Distributive laws

A∩(B∪C)=(A∩B)∪(A∩C) A∪(B∩C)=(A∪B)∩(A∪C)

Associative laws

A∪(B∪C)=(A∪B)∪C A∩(B∩C)=(A∩B)∩C

Idempotent laws

A∪A=A A∩A=A

Commutative laws

A∪B=B∪A A∩B=B∩A

Domination laws

A∪U=U A∩∅=∅

Identity laws

A∪∅=A A∩U=A

2 things to know about subsets

Because a ∈ ∅ is always false, ∅ ⊆ S for every set S. Because a ∈ S → a ∈ S, S ⊆ S, for every set S.

Conjunction (Rules of Inference for Propositional )

Corresponding Tautology: ((p) ∧ (q)) →(p ∧ q) Example: Let p be "I will study discrete math." Let q be "I will study English literature." "I will study discrete math." "I will study English literature." "Therefore, I will study discrete math and I will study English literature."

Resolution

Corresponding Tautology: ((¬p ∨ r ) ∧ (p ∨ q)) →(q ∨ r) ¬p ∨ r p ∨ q ________ ∴q ∨ r

Simplification (Rules of Inference for Propositional )

Corresponding Tautology: (p∧q) →p Let p be "I will study discrete math." Let q be "I will study English literature." "I will study discrete math and English literature" "Therefore, I will study discrete math."

Modus Tollens

Corresponding Tautology: (¬q∧(p →q))→¬p Let p be "it is snowing." Let q be "I will study discrete math." "If it is snowing, then I will study discrete math." "I will not study discrete math." "Therefore , it is not snowing."

Hypothetical Syllogism

Corresponding Tautology: ((p →q) ∧ (q→r))→(p→ r) Let p be "it snows." Let q be "I will study discrete math." Let r be "I will get an A." "If it snows, then I will study discrete math." "If I study discrete math, I will get an A." "Therefore , If it snows, I will get an A."

Disjunctive Syllogism

Corresponding Tautology: (¬p∧(p ∨q))→q Let p be "I will study discrete math." Let q be "I will study English literature." "I will study discrete math or I will study English literature." "I will not study discrete math." "Therefore , I will study English literature."

Addition (Rules of Inference for Propositional )

Corresponding Tautology: p →(p ∨q) p _________________ ∴p∨q Let p be "I will study discrete math." Let q be "I will visit Las Vegas." "I will study discrete math." "Therefore, I will study discrete math or I will visit Las Vegas."

Arithmetic Progression

Definition: A arithmetic progression is a sequence of the form: where the initial term a and the common difference d are real numbers.

Fibonacci Sequence

Definition: Define the Fibonacci sequence, f0 ,f1 ,f2,..., by: Initial Conditions: f0 = 0, f1 = 1 Recurrence Relation: fn = fn-1 + fn-2

Factorial Function

Definition: f: N → Z+ , denoted by f(n) = n! is the product of the first n positive integers when n is a nonnegative integer. f(6) = 6! = 1 ∙ 2 ∙ 3∙ 4∙ 5 ∙ 6 = 720

Natural Numbers

Denoted by N {0,1,2,3....}

Integers

Denoted by Z {...,-3,-2,-1,0,1,2,3,...}

True or false? ∅ = { ∅ }

False ∅ ≠ { ∅ }

Inclusive Or

For p ∨q to be true, either one or both of p and q must be true

Truth Sets of Quantifiers

Given a predicate P and a domain D, we define the truth set of P to be the set of elements in D for which P(x) is true. The truth set of P(x) is denoted by {x ∈ D|P(x)} ex: The truth set of P(x) where the domain is the integers and P(x) is "|x| = 1" is the set {-1,1}

Complement

If A is a set, then the complement of the A (with respect to U), denoted by Ā is the set U - A Ā = {x ∈ U | x ∉ A} ex: If U is the positive integers less than 100, what is the complement of {x | x > 70} Solution: {x | x ≤ 70}

Implication

If p denotes "I am at home." and q denotes "It is raining." then p →q denotes "If I am at home then it is raining

Biconditional

If p denotes "I am at home." and q denotes "It is raining." then p ↔q denotes "I am at home if and only if it is raining."

Conjunction

If p denotes "I am at home." and q denotes "It is raining." then p ∧q denotes "I am at home and it is raining."

Disjunction

If p denotes "I am at home." and q denotes "It is raining." then p ∨q denotes "I am at home or it is raining."

Negation

If p denotes "The earth is round.", then ¬p denotes "It is not the case that the earth is round," or more simply "The earth is not round."

Cardinality

If there are exactly n distinct elements in S where n is a nonnegative integer, we say that S is finite. Otherwise it is infinite. |ø| = 0 Let S be the letters of the English alphabet. Then |S| = 26 |{1,2,3}| = 3 |{ø}| = 1

Functions

Let A and B be nonempty sets. A function f from A to B, denoted f: A → B is an assignment of each element of A to exactly one element of B. We write f(a) = b if b is the unique element of B assigned by the function f to the element a of A.

Difference

Let A and B be sets. The difference of A and B, denoted by A - B, is the set containing the elements of A that are not in B. The difference of A and B is also called the complement of B with respect to A. A - B = {x | x ∈ A x ∉ B} = A ∩B *A-B would leave only the A (without the intersection) shaded*

Union

Let A and B be sets. The union of the sets A and B, denoted by A ∪ B, is the set {x|x ∈ A V x ∈ B} ex: What is {1,2,3} ∪ {3, 4, 5}? Solution: {1,2,3,4,5} *the whole venn diagram*

Russell's Paradox

Let S be the set of all sets which are not members of themselves. A paradox results from trying to answer the question "Is S a member of itself?" ex Henry is a barber who shaves all people who do not shave themselves. A paradox results from trying to answer the question "Does Henry shave himself?"

Inverse Functions

Let f be a bijection from A to B. Then the inverse of f, denoted f^(-1), is the function from B to A defined as f^(-1)(y)=x if f(x)=y

Composition

Let f: B → C, g: A → B. The composition of f with g, denoted f∘g(g) is the function from A to C defined by f∘g(x)=f(g(x))

Universal Instantiation

Our domain consists of all dogs and Fido is a dog. "All dogs are cuddly." "Therefore, Fido is cuddly."

Universal Generalization (UG)

P(c) for an arbitrary c ------------------ ∴∀xP(x)

Example of Set builder notation

S = {x | x is a positive integer less than 100} Q+ = {x ∈ R | x = p/q, for some positive integers p,q}

{{1,2,3},a, {b,c}} {*N*,*Z*,*Q*,*R*} This is an example of...

Sets being elements of sets.

Modus Ponens

Tautology (p ∧ (p →q)) → q Let p be "It is snowing." Let q be "I will study discrete math." "If it is snowing, then I will study discrete math." "It is snowing." "Therefore , I will study discrete math."

Cartesian Product

The Cartesian Product of two sets A and B, denoted by A × B is the set of ordered pairs (a,b) where a ∈ A and b ∈ B ex: A = {a,b} B = {1,2,3} A × B = {(a,1),(a,2),(a,3), (b,1),(b,2),(b,3)}

How is an empty set symbolized

The empty set is the set with no elements. Symbolized ∅, but {} also used

Intersection

The intersection of sets A and B, denoted by A ∩ B, is {x|x ∈ A ^ x ∈ B} ex: {1,2,3} ∩ {3,4,5} Solution: {3} {1,2,3} ∩ {4,5,6} ? Solution: ∅ *middle of venn diagram*

equality of ordered pairs

The ordered pairs (a,b) and (c,d) are equal if and only if a = c and b = d.

Subsets

The set A is a subset of B, if and only if every element of A is also an element of B. The notation A ⊆ B is used to indicate that A is a subset of the set B.

Power Sets

The set of all subsets of a set A, denoted P(A), is called the power set of A. ex: If A = {a,b} then P(A) = {ø, {a},{b},{a,b}}

Symmetric Difference

The symmetric difference of A and B, denoted by A ⊕ B is the set (A-B) U (B-A) ex: U = {0,1,2,3,4,5,6,7,8,9,10} A = {1,2,3,4,5} B ={4,5,6,7,8} What is A ⊕ B : Solution: {1,2,3,6,7,8} *everything but the middle of the venn diagram*

Exclusive Or

This is the meaning of Exclusive Or (Xor). In p ⊕ q , one of p and q must be true, but not both

Biconditional Statements

To prove a theorem that is a biconditional statement, that is, a statement of the form p ↔ q, we show that p → q and q →p are both true

Proof by Contradiction

To prove p, assume ¬p and derive a contradiction such as p ∧ ¬p. (an indirect form of proof). Since we have shown that ¬p →F is true , it follows that the contrapositive T→p also holds.

Showing that A is not a Subset of B

To show that A is not a subset of B, A ⊈ B, find an element x ∈ A with x ∉ B. (Such an x is a counterexample to the claim that x ∈ A implies x ∈ B.)

Showing that A is a Subset of B

To show that A ⊆ B, show that if x belongs to A, then x also belongs to B.

Set Equality

Two sets are equal if and only if they have the same elements. *no matter the order* {1,3,5} = {3, 5, 1} {1,5,5,5,3,3,1} = {1,3,5} are both equal sets

A tautology is...

a proposition which is always true

Absorption laws

a ∩ (a ∪ b)=a ∪ (a ∩ b)=a

Translate: You can access the Internet from campus only if you are a computer science major or you are not a freshman

a:You can access the internet from campus," c:"You are a computer science major," f: "You are a freshman." a→ (c ∨ ¬ f )

term of the sequence

a_n

codomain

are all the F(x)'s in F(x)=x

domain

are all the x's in F(x)=x

For interval notation name the symbol for *closed interval*

closed interval [a,b]

Asymmetric

if and only if (x,y) in R implies (y,x) not in R

Irreflexive

if and only if xRx does not hold for any x in A

reflexive

if and only if xRx for all x in A

Antisymmetric

if and only if xRy and yRx implies x = y

transitive

if and only if xRy and yRx implies xRz

symmetric

if and only xRy implies yRx

disjoint

if the intersection is empty, then A and B are said to be

image

is the *y*

Other names for functions

mappings or transformations.

For interval notation name the symbol for *open interval*

open interval (a,b)

one-to-one

or injective, if and only if f(a) = f(b) implies that a = b for all a and b in the domain of f. A function is said to be an injection if it is one-to-one. *passes the vertical line test*

onto

or surjective, if and only if for every element b∈B there is an element a∈A with f(a)=b . A function f is called a *surjection* if it is *onto.*

Translate: The automated reply cannot be sent when the file system is full

p: "The automated reply can be sent" q :"The file system is full." q→ ¬ p

converse of p →q and It raining is a sufficient condition for my not going to town

q →p If I do not go to town, then it is raining

universal set U

set containing everything currently under consideration

set of C

set of complex numbers.

set of Q

set of rational numbers

set of R

set of real numbers

Ceiling function

|-x-| is the smallest integer greater than or equal to x -1.5=-1

Negation (symbol)

¬

inverse of p → q and It raining is a sufficient condition for my not going to town

¬ p → ¬ q If it is not raining, then I will go to town.

contrapositive of p →q and It raining is a sufficient condition for my not going to town

¬q → ¬ p If I go to town, then it is not raining.

Biconditional (symbol)

↔

Universal Quantifier

∀xP(x) - P(x) holds for all x in the domain

Uniqueness Quantifier

∃!x P(x) means that P(x) is true for one and only one x in the universe of discourse.

Some student in this class has visited Mexico. Let M(x) denote "x has visited Mexico" and S(x) denote "x is a student in this class," and U be all people.

∃x (S(x) ∧ M(x))

Every student in this class has visited Canada or Mexico Let M(x) denote "x has visited Mexico" and S(x) denote "x is a student in this class," and U be all people Add C(x) denoting "x has visited Canada

∃x (S(x)→ (M(x)∨C(x)))

Existential Quantifier

∃xP(x) - P(x) holds for some x in the domain D

Conjunction (symbol)

∧

Disjunction (symbol)

∨