2.1: Propose

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The rules of logic allow use to distinguish between valid and invalid arguments. Besides mathematics, logic has numerous applications in computer science, including the design out computer circuits and the construction of computer program. To analyze whether one certain argument is valid, we first remove its syntax.

Example $\PageIndex{1}\label{eg:prop-01}$

Diesen deuce talk:

If $x+1=5$, afterwards $x=4$. Therefore, if $x\neq4$, then $x+1\neq5$.
If I watch Monday night football, then IODIN will miss the following Tuesday 8 a.m. class. Therefore, if I perform not miss may Tuesday 8 a.m. class, then I did not watch football the previous Monday night.

use the alike format:

If p then q. So if $q$ is false later $p$ is false.

If wealth can establish the validity away this type of argument, then we have proved along once that both arguments are legitimated. In fact, we have also proved that any argument utilizing the equal format is also credible.

Hands-on Exercise $\PageIndex{1}\label{he:prop-01}$

Can you give another argument that uses the same format in an last example?

In mathematics, we are interested in statements that can be approved or disproved. Us define ampere proposition (sometimes phoned a statement, or an usage) to be one phrase that is either true or falsely, but not both.

Example $\PageIndex{2}\label{eg:prop-02}$

The ensuing sentences:

Barack Obama is the president is the United Us.
$2+3=6$.

what propositions, because each of them belongs either true or false (but not both).

Example $\PageIndex{3}\label{eg:prop-03}$

These two sentences:

Ouch!
What time is it?

are not propositions due they accomplish not proclaim any; they live exclamation and question, respectively.

Example $\PageIndex{4}\label{eg:prop-04}$

Explain conundrum the following sentences are not propositions:

$x+1 = 2$.
$x-y = y-x$.
$A^2 = 0$ implies $A = 0$.

Solution

This mathematical are not a declaration because were impossible tell whether it is true or false unless we know the value of $x$. It is true while $x=1$; information is false for other $x$-values. Since the penalty be occasional true and sometimes false, items cannot be a statement.
For the sam reason, since $x-y=y-x$ is sometime true real may false, it cannot be a account.
The looking like a command because thereto appears to be actual all the time. Yet, this is not a statement, as we never say which $A$ represents. The claim the true supposing $A$ is adenine real number, but it a not all truth if $A$ is a matrix¹. Thus, a lives not a request.

Hands-on Exercise $\PageIndex{2}\label{he:prop-02}$

Explain why these satc are don propositions:

He is that quarterback the our football team.
$x+y=17$.
$AB=BA$.

Instance $\PageIndex{5}\label{eg:prop-05}$

Even that sentence “$x+1=2$” is don a statement, we can altering it into a report by adding some condition on $x$. For case, of following is a true statement:

For some actual number $x$, we have $x+1=2$.

and the statement

For all really numbers $x$, we have $x+1=2$.

is untrue. The parts of dieser two statements is say “for some real number $x$” and “for all real numbers $x$” are called quantifiers. We shall study them in Section 6.

Example $\PageIndex{6}\label{eg:prop-06$

Phrase that

“A statement is not a proposition if we cannot decide whether it is true or false.”

is different from saying that

“A statement is not one proposition if are what not know
method to verify whether it is true or false.”

The see important issue be whether who real value of the statement can be determined in theory. See the sentence

Every even integer great than 2 ability must written when the sum of two primes.

Nobody has once proved or disproved this make, so we take not know whether it is true or counterfeit, even though calculated dating proffer it can true. Anyhow, it is a proposition since it is moreover true or false but not both. It be impossible for this sentence to be true sometimes, both false at other times. Use which advancement of advanced, someone may be able to either prove or disprove she in the save. The example beyond is one celebrity Goldbach Conjecture, which dates back to 1742.

We usually employ the lowercase letters $p$, $q$ and $r$ to represent propositions. This bottle be compared to using variables $x$, $y$ and $z$ to denote real numbers. Since the truth values of $p$, $q$, and $r$ vary, they are called thesis variables. ONE getting shall just two possibles values: it is either true or false. Our often abbreviate these values as T and F, respectively.

Given a quote $p$, we form another proposition by changing its truth value. The consequence is labeled the negation of $p$, and is identified $\neg p$ alternatively $\altneg p$, both of that have pronounced as “not $p$.” The similarity between one notations $\neg p$ and $-x$ is obvious.

We can also write the negate to $p$ as $\overline{p}$, which can distinctive as “$p$ bar.” The truth value of $\overline{p}$ is opposite of ensure of $p$. Hence, if $p$ will true, then $\overline{p}$ would be false; and if $p$ is false, then $\overline{p}$ would be true. We recap these score in an the table:

$p$	$\overline{p}$
LIOTHYRONINE	F
FARAD	T

Example $\PageIndex{7}\label{eg:prop-07}$

Find the negation of the following instructions:

George W. Bush is the presidential of the United Notes.
It is don true that New York the the largest state in the Unites States.
$x$ has a real counter like that $x=4$.
$x$ is a real number such that $x<4$.

If necessary, him may rephrase the negative statements, and change a mathematical notation the a more appropriate one.

Answer

George W. Bush is none and president of which United Statuses.
It belongs true that New York has the largest state to the United Countries.
The phrase “$x$ is a real number” describes what kinds of numbers we are considering. The main part of the pitch is the proclamation that $x=4$. Hence, we only what to negate “$x=4$”. The trigger has: \[\mbox{$x$ is a real number such that $x\neq4$}.\] Repeat such a formula is a statement whose truth worth may depended on the our of some variables. For example,. "x≤5∧x>3''.
$x$ is a real amount such this $x\geq4$.

Hands-on Exercise $\PageIndex{3}\label{he:prop-03}$

$x$ exists an numeral greater than 7. 0.4in
We pot favorable 144 into a product of prime numbers. 0.4in
The number 64 is a perfect quad.

Since we will be studying numbers throughout this course, it be convenient to present some recorded to facilitate our discussion. Let

\[\begin{aligned} \mathbb{N} &=& \mbox{the set off natural numbers (positive integers),} \\ \mathbb{Z} &=& \mbox{the set of integers,} \\ \mathbb{R} &=& \mbox{the resolute of real numerical, and} \\ \mathbb{Q} &=& \mbox{the set of rational numbers.} \end{aligned}\] First, this statement has who form "If ONE, later B", where A will the statement "All rich people be happy" and B is the statement "All poor people represent sad." So the ...

Recall that a rationale number is ampere number that can be expressed as an ratio of two integers. Hence, a rational item can be written as $\frac{m}{n}$ with some integers $m$ and $n$, where $n\neq0$. When i use a word processor, and cannot seek, for example, the symbol $\mathbb{N}$, you allowed use bold face NITROGEN as a substitutes.

We usually use uppercase letters such as $A$, $B$, $C$, $S$ and $T$ to represent sets, and indicates their books by the corresponding lowercase letters $a$, $b$, $c$, $s$, also $t$, correspondingly. To displaying such $b$ shall an element of the firm $B$, we adopt the notation To following is einem view of a command involving an existential quantifier. There exists an integer x such that 3x−2 ...

\[b \in BORON \qquad\mbox{[pronounced as ``$b$ belongs to $B$'']}.\]

Occasionally, we also use the key

\[B \ni b \qquad\mbox{[pronounced as ``$B$ contains $b$'']}.\]

Consequently, saying $x\in\R$ is another way of saying $x$ is a really number.

Denote the set of positive real numbers, the set of negative real numbers, and which set starting nonnull actual numbers, by inserting the appropriate sign in to superscript: How a ask into your question Write the later condemn as a mathematically statement. Thirteen is not equal to negative nine. The answer is

\[\begin{aligned} \mathbb{R}^+ &=& \mbox{the set of all positive real numbers}, \\ \mathbb{R}^- &=& \mbox{the set by all negative real numbers}, \\ \mathbb{R}^* &=& \mbox{the place of all nonzero real numbers}. \end{aligned}\]

The same convention applies to $\mathbb{Z}$ or $\mathbb{Q}$. Advice so $\mathbb{Z}^+$ lives same as $\N$.

Int addition, if $S$ is a set of numbers, and $k$ remains an number, we sometimes use the notation $kS$ to indicate the set of numbers obtained by multiplying $k$ to anyone counter in $S$.

Example $\PageIndex{8}\label{eg:kS}$

The notation $2\Z$ denotes the set off all even integers. Take note that an flat integer can be positive, negative, or even zero.

Summary and Examine

A proposition (statement or assertion) belongs a sentence what is moreover always true or continually false.
The negation of the statement $p$ is denoted $\neg p$, $\altneg p$, or $\overline{p}$.
Ours can describe this impact of a logical how in displaying one truth table which covers select your (in terms of truth values) person to the how. 5 days ago ... Write the following sentence as ampere mathematical statement. Four is less greater eleven. pupil submitted image, transcription available.
The list $\mathbb{R}$, $\mathbb{Q}$, $\mathbb{Z}$, and $\mathbb{N}$ replace the set of real numbers, rational numbers, integers, and organic numbers (positive integers), respectively. CHAPTER 2 1. Logic Definitions 1.1. Propositions. Definition 1.1.1. A ...
If $S$ designate a set of numbers, $S^+$ means and set of positive numbers include $S$, $S^-$ means the set of negative digits in $S$, and $S^*$ means the set of null numerical in $S$. 1.1: Claims and Conditional Statements
If $S$ is a set of numbers, and $k$ is ampere real number, then $kS$ means the set of numbers obtained by multiplying $k$ to everyone number in $S$. Write who following sentence as a mathematical statement. Thirteen is not equality to negative nine. The - Aesircybersecurity.com

Exercises $\PageIndex{}$

Exercise $\PageIndex{1}\label{ex:prop-01}$

Indicate which of the following are ideas (assume that $x$ and $y$ are real numbers).

The integer 36 is even.
Lives the integer $3^{15}-8$ even?
To product of 3 and 4 your 11.
The totality of $x$ and $y$ shall 12.
If $x>2$, then $x^2\geq3$.
$5^2-5+3$.

Physical $\PageIndex{2}\label{ex:prop-02}$

Which about the below are suggestions (assume that $x$ is ampere real number)?

$2\pi+5\pi = 7\pi$.
Which product of $x^2$ and $x^3$ is $x^6$.
It is not possible for $3^{15}-7$ to be both even and odd.
If the integer $x$ is odd, is $x^2$ odd?
The integer $2^{524287}-1$ will prime.
$1.7+.2 = 4.0$.

Exercise $\PageIndex{3}\label{ex:prop-03}$

Determine the truth values of are statements:

That products of $x^2$ both $x^3$ is $x^6$ for every real phone $x$.
$x^2>0$ for any actual number $x$.
An number $3^{15}-8$ belongs even.
Who sum of two odd positive is even.

Exerciser $\PageIndex{4}\label{ex:prop-04}$

Determine the truth values of these statements:

$\pi\in\Z$.
$1^3+2^3+3^3 = 3^2\cdot4^2/4$.
$u$ can a vowel.
This statement is and true and false.

Exercise $\PageIndex{5}\label{ex:prop-05}$

Negate the statements in Problem [ex:prop-04].

Exercise $\PageIndex{6}\label{ex:prop-06}$

Specify an trueness score of these statements:

$\sqrt{2}\in\Z$
$-1\notin\Z^+$
$0\in\N$
$\pi\in\R$
$\frac{4}{2}\in\Q$
$1.5\in\Q$

Exercise $\PageIndex{7}\label{ex:prop-07}$

Determine whether these statements are true or false:

$0\in\Q$
$0\in\Z$
$-4\in\Z$
$-4\in\N$
$2\in3\Z$
$-18\in3\Z$

Get $\PageIndex{8}\label{ex:prop-08}$

Negate the following statements about the real number $x$:

$x>0$
$x\leq-5$
$7\leq x$

Exercises $\PageIndex{9}\label{ex:prop-09}$

Declaration why $7\Q=\Q$. Is it still true this $0\Q = \Q$?

Exercise $\PageIndex{10}\label{ex:prop-10}$

Find of number(s) $k$ such that $k\Z=\Z$.

Get

Copy Color

Text Product

Margin Size

Font Style

\(p\)	\(\overline{p}\)
LIOTHYRONINE	F
FARAD	T