Discrete Math

can always be itemized/counted and measurable

no gap in the contents of an object, continuous

the set of natural numbers without zero

set of natural numbers with zero

an unordered collection of distinct objects

each object in a set

What symbol is this: ∈

What are the two criteria for a set?

the set with no element

the number of elements in S

What does this denote: |S|

Is the set of real numbers continuous or discrete?

Is the set of natural numbers continuous or discrete?

A set A equals to another set B if and only if they have the same elements. ex. A = B

A set A is a ______ of another set B if and only if every element of A is also an element of B. Denoted A ⊆ B.

If A ⊆ B and A ≠ B, then A is a _______ _______ of B. Denoted A⊂B.

What are the two important properties of subsets?

2*|S| denotes what?

If A ⊆ B, B is called a __________ of A.

Let A and B be two sets. The _________ of A and B is the set of elements common to A and B. Denoted A∩B.

Let A and B be two sets. The ______ of A and B is the set of elements that belong to either A or B. Denoted A∪B.

holds for union and intersection: states that you can switch positions of operands (A&B). Denoted A∩B=B∩A

holds for union and intersection: no matter the order of operations, the result will be the same. Denoted (A∩B)∩C=A∩(B∩C)

Property that allows for the distribution of operations over addition or subtraction. Example: a(b + c) = ab + ac.

• holds for union and intersection

• C∩(A∪B) = (C∩A)∪(C∩B)

• C∪(A∩B) = (C∪A)∩(C∪B)

Let A be a set. The ________ of A is the set of all elements in the universal set which DO NOT belong to A. Denoted A' or A with a line over it.

What is the priority order of an expression with parentheses, union and intersection, and complement?

The cardinality of any set can be… (3 possibilities)

What are the two relationships De Morgan’s Law shows?

Let A and B be two sets. What is the set of all elements of A that DO NOT belong to B? What is the denotation?

Let A and B be two sets. What is the set of all ordered pairs of the format (a,b) that satisfies a belongs to A and b belongs to B? What is the denotation?

How do you find the cardinality of cartesian product?

(|AxB|) = ????

Let A be a set. What is the set of all subsets of A?

a declarative sentence that either true or false, but not both (no questions or undefined variables, fact, no opinions)

propositions that cannot be expressed in terms of simpler propositions (ex. Fanchao likes pizza.)

propositions are formed from existing propositions using logical operations (ex. If Fanchao is good at computer science, then he is good at programming.)

Let p be a proposition. What is the statement “It is not the case that p.”? What is the denotation?

given two propositions p and q, the __________ of p and q, denoted by p∧q. What is the truth table rule for this?

given two propositions p and q, the __________ of p and q, denoted by p∨q. What is the truth table rule for this?

Let p and q be two propositions. p and q are _________ if they have the same truth table value in EVERY possible case. What is the denotation?

Let p and q be two propositions. What is called the proposition that is true when exactly one of p and q is true and is false otherwise. What is the denotation?

Let p and q be two propositions. The ___________ statement is the proposition “if p, then q”. What is the truth table rule for this?

In a conditional statement, p→q, if p and q are compound propositions, then p→q is a ___________ _________ statement.

Given a conditional statements, p→q, the ___________ of this conditional statements is q→p.

Given a conditional statements, p→q, the inverse of this conditional statements is -p→-q.

Given a conditional statements, p→q, the ___________ of this conditional statement is -q→-p.

Let p and q be two propositions. The ______________ statement (p↔q) is the proposition “p if and only if q.” p↔q is true when p and q have the same truth value, and false otherwise.

a compound proposition that is always true (every case in the truth table is true)

a compound proposition that is always false (every case in the truth table is false)

a compound proposition that is neither a tautology nor a contradiction (most propositions are these)

a compound proposition is __________ if there is an assignment of truth variable (i.e. inputs) that makes it true

(some cases and true and some are false OR all cases are true)

a sequence of propositions (P1, P2,…,Pn, C.) The final proposition is called the conclusion. All the statements in this sentence except the conclusion are called premises.

(P1∧P2∧…∧Pn→C)

An argument is _______ if and only if all the premises being true forces the conclusion to be true. In other words, it is impossible for all the premises to be true and the conclusion to be false.

An argument is __________ if and only if it is valid and contains only true premises.

quantifier + variable + predicate

a symbol that represents a subject/object

∀ called the universal quantifier, means “for all,” and ∃, called the existential qualifier, means “there exists at least one”

a symbol represents a description of a subject/object or relation between subjects/objects

How do you negate statements?

∀ can distribute over a _______, but not over a ____________.

∃ can distribute over a _________, but not over a ________.

the set from which a variable takes values

ex. All students of MU like pizza. (______ is all students of MU.)

Whenever a variable is used, its ______ must be clarified.