ITSC 2175 Midterm
If an element is introduced for the first time in the proof, the definition is labeled ___
"Element definition" > must specify whether the element is arbitrary or particular.
(p → q) If it is raining today, the game will be cancelled. Converse:
( q → p ) > If the game is cancelled, it is raining today
(p → q) If it is raining today, the game will be cancelled. Inverse:
( ¬p → ¬q ) > If it is not raining today, the game will not be cancelled.
(p → q) If it is raining today, the game will be cancelled. Contrapositive:
( ¬q → ¬p ) > If the game is not cancelled, then it is not raining today.
Cartesian product
> denoted A x B > the set of all ordered pairs in which the first entry is in A and the second entry is in B > A x B = { (a, b) : a ∈ A and b ∈ B }
power set
> denoted P(A) > the set of all subsets of A > always contains an empty set
Boolean addition
> denoted by + > applies to two elements from {0, 1} and obeys the standard rules for addition, except for 1 + 1. > An outcome of 2 would not be allowed because all values in Boolean algebra must be 0 or 1 > The results of the addition operation are the same as the logical ∨ ("or") operation.
Boolean multiplication
> denoted by • > applies to two elements from {0, 1} and obeys the standard rules for multiplication > The results of the multiplication operation are the same as the logical ∧ ("and") operation.
complement
> denoted with a bar symbol, reverses that element's value. > Complementing a Boolean value is analogous to applying the ¬ ("not") operation in logic.
conditional operation
> denoted with the symbol →. > The proposition p → q is read "if p then q". >The proposition p → q is false if p is true and q is false, otherwise, p → q is true.
In p → q, the proposition p is called the ___, and the proposition q is called the ___
> hypothesis > conclusion
The empty set ∅ is ___ as { ∅ }
> not the same > The cardinality of { ∅ } is one since it contains exactly one element, which is the empty set
A statement with no free variables is a ___ because ___
> proposition > the statement's truth value can be determined
In the words of logic, the only way for a conditional statement to be false is if ___
> the hypothesis is true and the conclusion is false > If the hypothesis is false, then the conditional statement is true regardless of the truth value of the conclusion.
cardinality
> the number of elements in A > denoted by |A|
conjunctive normal form
A Boolean expression that is a product of sums of literals > Complement only applied to single variables. > No multiplication within a clause.
disjunctive normal form
A Boolean expression that is a sum of products of literals > Complement only applied to a single variable. > No addition within a term.
conditional proposition
A compound proposition that uses a conditional operation > "If there is a traffic jam today, then I will be late for work."
nested quantifiers
A logical expression with more than one quantifier that bind different variables in the same predicate Ex. (∀x ∃y Q(x, y))
quantified statement
A logical statement that includes a universal or existential quantifier > The quantifiers ∀ and ∃ are applied before the logical operations (∧, ∨, →, and ↔) used for propositions
predicate
A logical statement whose truth value is a function of one or more variables > P(5) is a proposition, but P(x) is a predicate
string
A sequence of characters
n-bit string
A string of length n
element
A value that can be plugged in for variable x
The Cartesian product of a set A with itself can be denoted as ___
A × A or A^2
literal
Boolean variable or the complement of a Boolean variable (for example, x or x).
The following argument is valid. p ∨ q p ----------- ∴ q (T/F)?
False
∃x∀y (x × y = 1) is the same as ∀x∃y (x × y = 1) (T/F)?
False
entry
The first entry of the ordered pair (x, y) is x and the second entry is y
elements
The objects in a set
alphabet
The set of characters used in a set of strings
equivalent
Two Boolean expressions are equivalent if they have the same value for every possible combination of values assigned to the variables contained in the expressions
disjoint
Two sets, A and B, are said to be disjoint if their intersection is empty (A ∩ B = ∅).
Boolean variables
Variables that can have a value of 1 or 0
bit
a character in a binary string
partition
a collection of non-empty subsets of A such that each element of A is in exactly one of the subsets For all i, Ai ⊆ A For all i, Ai ≠ ∅ A1, A2, ...,An are pairwise disjoint. A = A1 ∪ A2 ∪ ... ∪ An
set
a collection of objects
rational number
a number that can be expressed as the ratio of two integers in which the denominator is non-zero
minterm
a product of literals that must contain every literal in the function
existentially quantified statement
a proposition that uses the existential quantifier (∃) to claim that "There exists an x, such that P(x)" which asserts that P(x) is true for at least one possible value for x in its domain > ∃x P(x)
universally quantified statement
a proposition that uses the universal quantifier (∀) to claim that "for all x, P(x)" which asserts that P(x) is true for every possible value for x in its domain > ∀x P(x)
irrational number
a real number that is not rational
argument
a sequence of propositions, called hypotheses, followed by a final proposition, called the conclusion
logical proof
a sequence of steps, each of which consists of a proposition and a justification. If the proposition in a step is a hypothesis, the justification is "Hypothesis". Otherwise, the proposition must follow from previous steps by applying one law of logic or rule of inference > The justification indicates which rule or law is used and the previous steps to which it is applied.
set builder notation
a set is defined by specifying that the set includes all elements in a larger set that also satisfy certain conditions. Ex. A = { x ∈ S : P(x) }
Boolean algebra
a set of rules and operations for working with variables whose values are either 0 or 1
theorem
a statement that can be proven to be true
proposition
a statement that is either true or false
binary string
a string whose alphabet is {0, 1}
truth value
a value indicating whether the proposition is actually true or false
bound variable
a variable that is bound to a quantifier > ∀x P(x)
free variable
a variable that is free to take on any value in the domain > variable x in the predicate P(x)
proof
consists of a series of steps, each of which follows logically from assumptions, or from previously proven statements, whose final step should result in the statement of the theorem being proven
compound proposition
created by the conjunction of individual propositions with logical operations
exclusive or
evaluates to true when p is true and q is false or when q is true and p is false > denoted with ⊕
particular element
may have properties that are not shared by all the elements of the domain
biconditional operation
p if and only if q > The proposition p ↔ q is true when p and q have the same truth value and is false when p and q have different truth values. > denoted p ↔ q
I will share my cookie with you only if you share your soda with me. (p → q or q → p)?
p → q > p only if q.
If two propositions are logically equivalent, then one can be ___
substituted for the other within a more complex proposition.
An argument is valid if ___ otherwise the argument is invalid
the conclusion is true whenever the hypotheses are all true
direct proof
the hypothesis p is assumed to be true and the conclusion c is proven as a direct result of the assumption
The use of parentheses ( ) for an ordered pair indicates ___
the order of entries is significant
The use of parentheses { } for an ordered pair indicates ___
the order of entries isn't significant
combinatorial circuit
the output of the circuit depends only on the present combination of input values and not on the state of a circuit > can not store information over time > used to store a single bit of information
complement
the set of all elements in U that are not elements of A > denoted Ā
union
the set of all elements that are elements of A or B > denoted A ∪ B
Z
the set of all integers.
R- and Z- are:
the set of all negative values in their respective sets
R+ and Z+ are:
the set of all positive values in their respective sets
domain
the set of all possible values for the variable > all values, both true and false
symmetric difference
the set of elements that are a member of exactly one of A and B, but not both > denoted A ⊕ B
difference
the set of elements that are in A but not in B > denoted A - B
N
the set of natural numbers, which includes all integers greater than or equal to 0.
Q
the set of rational numbers, which includes all real numbers that can be expressed as a/b, where a and b are integers and b ≠ 0.
R
the set of real numbers
Logic
the study of formal reasoning
empty string
the unique string whose length is 0 and is usually denoted by the symbol λ
conjuction
using a logical operator such as 'and' to create a compound proposition from individual propositions > only true when both propositions are true > denoted with ^
disjunction
using a logical operator such as 'or' (inclusive) to create a compound proposition from individual propositions > true when either proposition is true > denoted with v
cardinality is denoted by ___
|A|
De Morgan's law for quantified statements
¬∀x P(x) ≡ ∃x ¬P(x) ¬∃x P(x) ≡ ∀x ¬P(x)
De Morgan's laws for nested quantified statements:
¬∃x ∀y P(x, y) ≡ ∀x ∃y ¬P(x, y) ¬∃x ∀y (P(x) ∨ ¬Q(y)) ≡ ∀x ∃y ¬(P(x) ∨ ¬Q(y))
Existential instantiation
∃x P(x) ----------- ∴ (c is a particular element) ∧ P(c)
ordered triple
> denoted (x, y, z) > An ordered list of three items
logical operation
combines propositions using a particular composition rule
OR gate
computes Boolean addition
AND gate
computes Boolean multiplication
inverter
computes the complement
ordered pair
(x, y)
to reduce a complex compound proposition:
1. Use DM's laws to distribute negations 2. Look through the laws to see which can apply to the proposition 3. Try to look forward with each possible option to see if any of the results are recognizable as being closer to the desired result 4. If simplifying alone doesn't help, try expanding (use the laws backwards) 5. Repeat steps 1-3, trying to get the expanded form reduced into the desired result
order of operations:
1. ¬ (not) 2. ∧ (and) 3. ∨ (or)
universal set
> a set that contains all elements mentioned in a particular context > denoted by the variable U
Venn diagrams
> A rectangle is used to denote the universal set U, and oval shapes are used to denote sets within U. > Venn diagrams can indicate which specific elements are inside and outside the set. > An element is drawn inside the oval if it is in the set represented by the oval.
Rules of inference
> Modus ponens > Modus tollens > Addition > Simplification > Conjunction > Hypothetical syllogism > Disjunctive syllogism > Resolution
empty set (null set)
> The set with no elements > denoted by the symbol ∅
proper subset
If A ⊆ B and there is an element of B that is not an element of A (i.e., A ≠ B), then A is a proper subset of B > denoted as A ⊂ B
subset
If every element in A is also an element of B, then A is a subset of B > denoted as A ⊆ B
concatenation
If s and t are two strings, then the concatenation of s and t (denoted st) is a longer string obtained by putting s and t together
clause
In conjunctive normal form, each term in the product that is a sum of literals
the representation of a boolean function as a sum of minterms includes:
all minterms which result in 1 when plugged into the function the terms in the minterm are multiplied together then all the minterms are added together to represent the function
counterexample
an assignment of values to variables that shows that a universal statement is false
counterexample
an element in the domain for which the predicate is false > A single counterexample is sufficient to show that a universally quantified statement is false
set identity
an equation involving sets that is true regardless of the contents of the sets in the expression
ordered n-tuple
an ordered list of n items
Universal generalization
c is an arbitrary element P(c) ----------- ∴ ∀x P(x)
Existential generalization
c is an element (arbitrary or particular) P(c) ----------- ∴ ∃x P(x)
Universal instantiation
c is an element (arbitrary or particular) ∀x P(x) ----------- ∴ P(c)
Boolean expressions
can be built up by applying Boolean operations to Boolean variables or the constants 1 or 0.
odd integer
can be expressed as 2k + 1 for some integer k
even integer
can be expressed as 2k for some integer k
roster notation
definition of a set is a list of the elements enclosed in curly braces with the individual elements separated by commas Ex. A = { 2, 4, 6, 10 }
reordering the hypotheses ___ change whether an argument is valid or not
does not > two arguments are considered to be the same even if the hypotheses appear in a different order
finite set
has a finite number of elements
infinite set
has an infinite number of elements
arbitrary element
has no special properties other than those shared by all the elements of the domain
truth table
have a row for every possible combination of truth assignments for the statement's variables > If there are n variables, there are 2^n rows
pairwise disjoint
if every pair of distinct sets in the sequence is disjoint
contradiction
if the proposition is always false, regardless of the truth value of the individual propositions that occur in it
tautology
if the proposition is always true, regardless of the truth value of the individual propositions that occur in it
logically equivalent
if two compound propositions have the same truth value regardless of the truth values of their individual propositions > notated by (s ≡ r) > Propositions s and r are logically equivalent if and only if the proposition s ↔ r is a tautology
non-negative
if x ≥ 0
form of an argument
is expressed in English and is obtained by replacing each individual proposition with a variable
intersection
is the set of all elements that are elements of both A and B > denoted A ∩ B
De Morgan's laws
logical equivalences that show how to correctly distribute a negation operation inside a parenthesized expression >The two versions of De Morgan's laws are: >> ¬(p ∨ q) ≡ (¬p ∧ ¬q) >> ¬(p ∧ q) ≡ (¬p ∨ ¬q)
an argument is denoted as
p1 p2 .... pn ----- ∴ c > p1 ... pn are the hypotheses > c is the conclusion > The symbol ∴ reads "therefore"
proof by contrapositive
proves a conditional theorem of the form p → c by showing that the contrapositive ¬c → ¬p is true > ¬c is assumed to be true and ¬p is proven as a result of ¬c
proof by exhaustion
proving that a statement by checking each element individually
gates
receives some number of Boolean input values and produces an output based on the values of the inputs > a gate implements a simple Boolean function
existential/universal instantiation
replace a quantified variable with an element of the domain
existential/universal generalization
replace an element of the domain with a quantified variable
negation
reverses the truth value of a proposition > denoted with -
input/output table
shows the output value of the function for every possible combination of input values
axioms
statements assumed to be true