Cs 214 final study

What does the set defined by A - B have in it?

A - B would result in the set of elements that are not common between them. for example, if A = {a, b, c, d}, and B was {c, d, e, f}, then A - B would be {a, b, e, f}

what is a bijective function?

A bijective function is a type of function that is both injective (one-to-one) and surjective (onto). Each element in the domain maps to a unique element in the codomain, and every element in the codomain is mapped to by exactly one element in the domain. Bijective functions establish a one-to-one correspondence between the domain and the codomain, ensuring that the mapping is both unique and complete.

What is a contradiction?

A contradiction is a statement in propositional logic that is always false, regardless of the truth values of its individual components. In other words, a contradiction is a logical expression that evaluates to false under all possible truth value assignments to its variables. Contradictions represent statements that cannot be true simultaneously and are used to identify inconsistencies in logical arguments.

What is a function?

A function is a relation between a set of inputs (the domain) and a set of possible outputs (the codomain) where each input is related to exactly one output. In simpler terms, it assigns each input value to exactly one output value.

What is a one-to-one function?

A one-to-one function, also known as an injective function, is a type of function where each element in the domain maps to a distinct element in the codomain. In simpler terms, no two different elements in the domain map to the same element in the codomain. One-to-one functions pass the horizontal line test, meaning that no horizontal line intersects the graph of the function more than once.

What is a tautology?

A tautology is a statement in propositional logic that is always true, regardless of the truth values of its individual components. In other words, a tautology is a logical expression that evaluates to true under all possible truth value assignments to its variables

Real numbers (R)

All numbers that are real, this does not include stuff like i and 0+i

what is an onto function?

An onto function, also known as a surjective function, is a type of function where every element in the codomain is mapped to by at least one element in the domain. In other words, for every element in the codomain, there is at least one element in the domain that maps to it. Onto functions ensure that no element in the codomain is left unmapped. They cover the entire range of the codomain.

To make the truth table for a statement, you:

Do the true false's for all variables, mini combinations, big combinations, and full statement

the negation of P' is:

P

Negation of P->Q is:

P ^ Q'

the negation of P is:

P'

the negation of PvQ is:

P'^Q'

the negation of P^Q is:

P'vQ'

PROCEDURE When asked to prove or validate by wffs, or do something using wffs, what are you being asked to do?

Proofs, like what we did with the rules we have (as we have above) and using them and our proofs like we've been taught.

∃ means:

The existential quantifier is symbolized by ∃, and is read "there exists one," "for at least one," or "for some." Thus the expression (∃ x)(x > 0) is read "there exists an x such that x is greater than zero."

PROCEDURE What is the inductive process for PMI 2?

The inductive process for PMI 2 is similar to PMI 1, but with some differences. the First Principle Involves proving a base case and then showing that if the statement holds for an arbitrary integer k, it also holds for k+1. the Second Principle Assumes the statement holds for ALL INTEGERS UP TO a certain value k and then proves it for the next integer k+1 The second principle is often used when the base case is not the smallest integer, making it a more flexible approach in some cases.

what is the preimage of the function?

The preimage of a function refers to the set of all elements in the domain that map to a specific element in the codomain. In simpler terms, given a function f, if y is an element in the codomain, the preimage of y consists of all elements in the domain that map to y under the function f. The preimage helps identify all the possible inputs that lead to a particular output in the function.

what is the range of a function?

The range of a function is the set of all possible output values (or elements) that the function can produce when all possible inputs are considered. In simpler terms, the range is the set of all values that the function can attain.

Upside down A means:

The universal quantifier is symbolized by an upside down A, , and is read "for all," "for every," "for each," or "for any." Thus the example sentence, "For every x, x > 0", can be symbolized by (upsd(A)x)(x > 0)

When you want a set to be reflexive, what does that mean?

When all elements in the domain are mapped to itself, so a reflexive set with domain {1,2,3} can be something like: {(1,1) (1,2) (2,2) (2,3) (3,3)} Because 1,2, and 3 are all mapped to itself

PROCEDURE Proofs: Here's a quick refresher on how to go about doing proofs:

You are trying to show, trying to get the right side of the implication statement from the left side, using the hypothesis, and rules and things we have above. Example: A' ^ (B -> A) -> B' 1: A' : Hyp 2: B→A : Hyp 3: B' : 1,2,MT

if you want a set to be antisymmetric, what does that mean?

a set is antisymmetric, if for every unique number x within, there is a pair x,x in the set, AND if there are no symmetric x,y y,x pairs in the set

from P, we can derive P v Q

addition

Naturals (N)

all positive integers, including 0

what is the power set, (that funky p) pwr(A) mean?

all possible combinations of the elements in the set, including the single sets, with just the individual elements, the double sets, with all combinations of two elements, and so forth, as well as the none set

Integers (Z)

all whole integers, positive and negative

PROCEDURE if the range of what you are trying to prove is finite, what's a possibly easier way to prove or disprove it?

an Exhaustive Proof, ie, showing that its true for every element in the range, or finding a counter example

Basic negation

as a basic concept, you are putting the whole statement in parenthesis, and adding the negation symbol, if its easier for you to understand it in non english representation

(PvQ)vR can change to Pv(QvR) (P^Q)^R can change to (P^(Q^R)

associative(asso)

How do you solve, (get the closed form of) a recursion statement?

basic recursion, 'expand guess verify' sequence.: you take the recursion statement you have, and you expand it, meaning you take your statement, and replace the P(n-1) with the statement, only with P(n-2), then, you replace just the P(n-2) in that, and replace it with the entire statement, but with P(n-3) then, at that point, you can probably look for the pattern, and get it at what you GUESS is the k form of the recursion. then, to get a good idea, you replace k with n-1, and where you'll have P(n-(where you had k)(n-1), you can replace with P(1), which you'll probably have a definition for, and you simplify the resulting formula, and voila, you have your closed form recursion definition

from PvQ, can change to QvP from P^Q, can change to Q^P

commutative (comm)

PvP' can get 1 P^P' can get 0

complement

from P, Q, we can derive P^Q

conjunction(con)

from P->Q, we can derive Q' -> P' from Q'->P', we can derive P->Q

contraposition(cont)

(PvQ)' can change to P'^Q' (P^Q)' can change to P'vQ'

de morgans laws (de morg)

from P <-> Q can change to (P->Q) ^ (Q->P)

definition of equivalence(equ)

from PvQ, P', we can derive Q

disjunctive syllogism (ds)

from P^(QvR) we can derive (P^Q) v (P^R) from Pv(Q^R) we can derive (PvQ)^(PvR)

distributive(dist)

from P can change to (P')'

double negation(dn)

∈ means:

element of, if x ∈ A, then x is an element of A

from (P^Q)->R, we can derive P->(Q->R)

exportation(exp)

When you want a set to be symmetric, what does that mean?

for every pair {x,y} in the set, there needs to be a corresponding {y,x}

negation example

for example, Peter is tall and thin, can be written as A^T the negation would be (A^T)' (it is false that Peter is tall and thin) or simplified further, A' v T' (peter is not tall, OR he is not thin. you would NOT get A'^T'

for binary relations x p y, p on some thing, (often Z, for integers), with <-> (some algebraic statement, ie, x^2 + y^2 = z^2), what do each of these parts here represent?

for the most part, this is talking about whether the binary relation between two numbers satisfies a given condition or not. All the numbers and how its phrased make it seem confusing, but its actually pretty simple. lets take an example from the bonus assignment to show you: x p y <-> x^2 + y^2 = z^2 for some integer z; (1,0), (3,9), (2,2), (-3,4) so, for each of the pairs (v,w) above, the number on the left is x, and the number on the right is y. the question is asking, which of the pairs are x and y pairs that satisfy the conditions, and thus belong to p? (1,0): 1^2 + 0^2 = 1^2, which is a squared integer. (3,9): 3^2 + 9^2 = 90, which is not a squared integer, it does not belong to p (2,2) 2^2 + 2^2 = 8, which is not an integer squared and does not belong to p. (-3,4) -3^2 + 4^2 = 25, which is 5 squared, and belongs to p

from P->Q, Q->R, we can derive P->R

hypothetical syllogism (hs)

Pv0 can get P P^1 can get P

identity

With recursion relations that are geometric sequences, there's actually a pretty straightforward approach and easy tip to solving them

if a_n = D*A(n-1), and A_0 = k, then A_n = k*(D^n)

when you want a set to be transitive, what does that mean?

if you have a domain of w x y and z and your set is {(w,y) (y,w) (w,w) (y,y)} its transitive because w maps to y, and y maps to w, so you need a w,w and y maps to w, and w maps to y, so you need a y,y thus, just the set {{w,w}} would be transitive, and antisymmetric

from P->Q can change to P'vQ

implication(imp)

Rational Numbers (Q)

includes things like decimals and fractions, does not include things like pi

from P, P', we can derive Q

inconsistency

⊂

is a proper subset of. if B is a proper subset of A, then all elements in B are in A, BUT there are elements in A that are not in B

PROCEDURE when you want to find the closure, (for some reflexive, antisymmetric, etc) of a set, what does that mean to do?

it means to add the set pairs (x,y) and such that would make the set you are working with into a reflexive, or transitive, etc. set

Prove the following tautologies by starting with the left side and finding a series of equivalent wffs that will convert the left side into the right side. what does this mean to do?

its like simplification, you take the basic rules you know how to do, like double negation, B v B = B, and such to simplify the left side into what you have on the right.

from P, P->Q, we can derive Q

modus ponenes (mp)

from P->Q, Q', we can derive P'

modus tollens(mt)

from P, we can derive P^P from PvP, we can derive P

self reference(self)

from P^Q, we can derive P, Q

simplification (sim)

⊆

subset. if B is a subset of A, then all elements in B are in A.

from x (being true), we can also logically say, y is true

the following meaning of the rules format

what is the domain of a function?

the set of inputs the function has

what is the co-domain of the function?

the set of outputs

⋂

this is the intersection. Its another thing to do with sets, and A ⋂ B means the set of all elements that A and B have in common. Similar in function in equations to an AND. behaves like multiplications

⋃

this is the union. its another thing to do with sets, and A ⋃ B means all elements in both. Similar in function in Equations to an OR. behaves like addition

with sets, B X A means the Cartesian product of the set. What does this mean?

this means creating ordered pairs where the first element comes from set B and the second element comes from set C. For example, the pair (1,2) means 1 is from set B and 2 is from set C. The result is a set of all possible ordered pairs that can be formed from elements in B and C. example: Let D={a,b} and E={1,2,3}. Find D×E. D×E={(a,1),(a,2),(a,3),(b,1),(b,2),(b,3)}

PROCEDURE how do you do a proof by contradiction?

to do a proof by contradiction, you take what you have, and set out to prove that it does the opposite of what it says its gonna do, then get it to the point where you show that it can never do the opposite, thus, proving that it is true by contradiction

PROCEDURE if you are trying to prove the validity of an argument, it will typically have a compound statement on the right side of the implication ie, more than one variable

to do this, you'll need to use Deduction(ded) as one of your hyps, to get the first part of that statement, and change what you NTS to just be the second part, then prove the statement, using that first part as one of your hyps, thus proving that its possible to get second part, using first part.

can a set have more than one property? (symmetric, transitive, etc.)

yes, and you will likely be asked to test for multiple. for example, the set {{w,w}} has all four properties

thus, to negate:

you negate each element and connective within the statement

PROCEDURE What is the inductive process for PMI 1?

you take the base case, showing what the thing would be for F(1), then you use the Inductive hypothesis, and say its true for for an arbitrary k, then do the inductive step to prove its true for k+1 too, subbing in your formula for k as the first part of the series