Prove asymptotic upper and lower hounds for each of the following specified otherwise, assume that in each case, T(n) = 1 (or any small constant) for small value You may assume that n = c^k for some constant c that you choose. Make your bounds as tight as (No need to specify the origin of your guess.) T(n0 = 8T(n/3) + n^1.83838383... T(n) = T(n - 1) = 1/n T(n) = 16T(n/2) + (n log n)^4. T(n) = 2T(n/2) + n/lg n. T(n) = T(n - 1) + T(n - 2) + 1 with base case of T(1) = 1 and T(2) = 2 Solution A statement are often outlined as a declaratory sentence, or a part of a sentence, that\'s capable of getting a truth-value, like being true or false. So, as an example, the subsequent area unit statements: George W. Bush is that the forty third President of the us. Paris is that the capital of France. Everyone born on Monday has purple hair. Sometimes, a press release will contain one or a lot of alternative statements as elements. contemplate as an example, the subsequent statement: Either Ganymede may be a moon of Jupiter or Ganymede may be a moon of Saturn. While the on top of sentence is itself a press release, as a result of it\'s true, the 2 elements, \"Ganymede may be a moon of Jupiter\" and \"Ganymede may be a moon of Saturn\", area unit themselves statements, as a result of the primary is true and therefore the second is fake. The term proposition is typically used synonymously with statement. However, it\'s typically accustomed name one thing abstract that 2 totally different statements with an equivalent which means area unit each aforementioned to \"express\". during this usage, nation sentence, \"It is raining\", and therefore the French sentence \"Il pleut\", would be thought-about to specific an equivalent proposition; equally, the 2 English sentences, \"Callisto orbits Jupiter\" and \"Jupiter is orbitted by Callisto\" would even be thought-about to specific an equivalent proposition. However, the character or existence of propositions as abstract meanings continues to be a matter of philosophical dispute, and for the needs of this text, the phrases \"statement\" and \"proposition\" area unit used interchangeably. Propositional logic, conjointly referred to as linguistic string logic, is that branch of logic that studies ways that of mixing or neutering statements or propositions to create a lot of difficult statements or propositions. change of integrity 2 easier propositions with the word \"and\" is one common approach of mixing statements. once 2 statements area unit joined along side \"and\", the advanced statement fashioned by them is true if and as long as each the element statements area unit true. owing to this, associate argument of the subsequent kind is logically valid: Paris is that the capital of France and Paris contains a population of over 2 million. Therefore, Paris contains a population of over 2 million. Propositional logic for the most part involves learning logical connectives like the words \"and\" and \"or\" and therefo.

1. Prove asymptotic upper and lower hounds for each of the following specified otherwise, assume
that in each case, T(n) = 1 (or any small constant) for small value You may assume that n = c^k
for some constant c that you choose. Make your bounds as tight as (No need to specify the
origin of your guess.) T(n0 = 8T(n/3) + n^1.83838383... T(n) = T(n - 1) = 1/n T(n) = 16T(n/2)
+ (n log n)^4. T(n) = 2T(n/2) + n/lg n. T(n) = T(n - 1) + T(n - 2) + 1 with base case of T(1) = 1
and T(2) = 2
Solution
A statement are often outlined as a declaratory sentence, or a part of a sentence, that's capable of
getting a truth-value, like being true or false. So, as an example, the subsequent area unit
statements:
George W. Bush is that the forty third President of the us.
Paris is that the capital of France.
Everyone born on Monday has purple hair.
Sometimes, a press release will contain one or a lot of alternative statements as elements.
contemplate as an example, the subsequent statement:
Either Ganymede may be a moon of Jupiter or Ganymede may be a moon of Saturn.
While the on top of sentence is itself a press release, as a result of it's true, the 2 elements,
"Ganymede may be a moon of Jupiter" and "Ganymede may be a moon of Saturn", area unit
themselves statements, as a result of the primary is true and therefore the second is fake.
The term proposition is typically used synonymously with statement. However, it's typically
accustomed name one thing abstract that 2 totally different statements with an equivalent which
means area unit each aforementioned to "express". during this usage, nation sentence, "It is
raining", and therefore the French sentence "Il pleut", would be thought-about to specific an
equivalent proposition; equally, the 2 English sentences, "Callisto orbits Jupiter" and "Jupiter
is orbitted by Callisto" would even be thought-about to specific an equivalent proposition.
However, the character or existence of propositions as abstract meanings continues to be a matter
of philosophical dispute, and for the needs of this text, the phrases "statement" and
"proposition" area unit used interchangeably.
Propositional logic, conjointly referred to as linguistic string logic, is that branch of logic that
studies ways that of mixing or neutering statements or propositions to create a lot of difficult
statements or propositions. change of integrity 2 easier propositions with the word "and" is one
common approach of mixing statements. once 2 statements area unit joined along side "and",
the advanced statement fashioned by them is true if and as long as each the element statements
area unit true. owing to this, associate argument of the subsequent kind is logically valid:

2. Paris is that the capital of France and Paris contains a population of over 2 million.
Therefore, Paris contains a population of over 2 million.
Propositional logic for the most part involves learning logical connectives like the words "and"
and "or" and therefore the rules crucial the truth-values of the propositions they're accustomed
be part of, still as what these rules mean for the validity of arguments, and such logical
relationships between statements as being consistent or inconsistent with each other, still as
logical properties of propositions, like being tautologically true, being contingent, and being self-
contradictory. (These notions area unit outlined below.)
Propositional logic conjointly studies approach of modifying statements, like the addition of the
word "not" that's accustomed modification associate affirmative statement into a negative
statement. Here, the basic logical principle concerned is that if a given affirmative statement is
true, the negation of that statement is fake, and if a given affirmative statement is fake, the
negation of that statement is true.
What is distinctive regarding mathematical logic as opposition alternative (typically a lot of
complicated) branches of logic is that mathematical logic doesn't influence logical relationships
and properties that involve the elements of a press release smaller than the easy statements
creating it up. Therefore, mathematical logic doesn't study those logical characteristics of the
propositions below in virtue of that they represent a legitimate argument:
George W. Bush may be a president of the us.
George W. Bush may be a son of a president of the us.
Therefore, there's somebody WHO is each a president of the us and a son of a president of the
us.
The recognition that the on top of argument is valid needs one to acknowledge that the topic
within the 1st premise is that the same because the subject within the second premise. However,
in mathematical logic, straightforward statements area unit thought-about as indivisible wholes,
and people logical relationships and properties that involve elements of statements like their
subjects and predicates don't seem to be taken into thought.
Propositional logic are often thought of as primarily the study of logical operators. A logical
operator is any word or phrase used either to switch one statement to form a distinct statement, or
be part of multiple statements along to create a a lot of difficult statement. In English, words like
"and", "or", "not", "if ... then...", "because", and "necessarily", area unit all operators.
A logical operator is alleged to be truth-functional if the truth-values (the truth or falsity, etc.) of
the statements it's accustomed construct forever rely entirely on the reality or falsity of the
statements from that they're created. nation words "and", "or" and "not" area unit (at least
arguably) truth-functional, as a result of a compound statement joined along side the word
"and" is true if each the statements thus joined area unit true, and false if either or each area unit

3. false, a compound statement joined along side the word "or" is true if a minimum of one among
the joined statements is true, and false if each joined statements area unit false, and therefore the
negation of a press release is true if and as long as the statement negated is fake.
Some logical operators don't seem to be truth-functional. One example of associate operator in
English that's not truth-functional is that the word "necessarily". whether or not a press release
fashioned mistreatment this operator is true or false doesn't rely entirely on the reality or falsity
of the statement to that the operator is applied. as an example, each of the subsequent statements
area unit true:
2 + 2 = 4.
Someone is reading an editorial in a very philosophy book of facts.
However, allow us to currently contemplate the corresponding statements changed with the
operator "necessarily":
Necessarily, 2 + 2 = 4.
Necessarily, somebody is reading an editorial in a very philosophy book of facts.
Here, the primary example is true however the second example is fake. Hence, the reality or
falsity of a press release mistreatment the operator "necessarily" doesn't rely entirely on the
reality or falsity of the statement changed.