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in this video i’ll show you how to use logical connectives and you’ll learn the

seven logical connectives that you must master

of course for at least thousands of years people have been developing logic

but did you know that even in the last 150 years propositional logic continues

to make serious advancements so let’s learn what logical connectives are

hi everyone welcome back i’m dave in this video seven logical connectives

that you must master is part of the series logic and mathematical proof

in-depth tutorials for beginners i recommend checking out the series the

link below is in the description so let’s go ahead and get started so

first i’m going to talk about mathematical statements and then we’re

going to talk about the seven logical connectives the negation connective and

then we’ll talk about the conjunction and disjunction connectives

then we’ll talk about the implication and equivalence connections

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connectives and then we’ll talk about the alternative and joint negation

connectives so let’s go first up is mathematical statements what are they

so in other words in order to get started here

we’re going to need to know what we’re connecting together

so we’re going to be connecting together mathematical statements

and so the seven connectives that we’re going to cover are negation

and the disjunction the or and the conjunction which is the and so we’re

going to be using these symbols right here on the right hand on the left hand

side the right arrow for implies and the left and right arrow for if and only if

and then we’re going to be looking at the up arrow or the alternative negation

and the down arrow which represent the nor so we’re going to be looking at these

seven um logical connectives in this video and so you know up first is what is a

mathematical statement so a mathematical statement is

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a declarative sentence that can be classified as either true or false

so for example something like one plus two equals three

or you could have a one plus two equals four

so these are examples of mathematical statements

here on this mathematical statement in you know it’s a declarative sentence

so we’re declaring that the left-hand side is equal to the right-hand side

and over here on this one we’re declaring that the left-hand side is

equal to the right-hand side now this is a mathematical statement it’s a

declarative sentence it’s either true or false but not both

and in fact we know this one is true and in fact we know this

statement right here is false so these are mathematical statements

and now there’s going to be two types of mathematical statements the compound

statements and the simple statements in fact all comp all statements all

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mathematical statements will label us all of them will be compound statements

and we’re going to form lots of compound statements using logical connectives

but the ones the mathematical statements that do not contain in these seven

connectives the seven connectors that i just listed

so if it doesn’t contain any of them then it’s called a simple statement so

these two are simple statements here these are simple statements

so we can make compound statements using simple statements we can make more

compound statements so for example if i call this statement right here a p

and i call the statement right here q just to label them then i can do

something like form the logical connective

using the logical connective and i can form the statement p and q

and so that will represent one plus two equals three and

one plus two equals four so this will be a compound statement

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these are also compound statements because they’re simple statements

but these simple statements are not compound statements

these are compound statements sorry i said that wrong these simple

statements are also compound statements and this is the compound statement right

here we can do another one for example or so we can do one plus two equals three

or one plus two equals four so these are these these are all compound statements

and these are simple statements right here

there’s no logical connectives in these statements here

so using logical connectives we can create a lot more uh statements so

we’re going to use the convention here of a lowercase p

for propositional variables and then uppercase p for compound statements um so

that’s just going to kind of make things a little bit easier

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um so we’re going to define compound statements as follows we’re going to say

all simple statements are also compound statements

and then we’re going to say well we can use our connectives to construct more

compound statements so you can take any p and q

any compound statements p and q and you can build more compound statements so

nothing else is a compound statement except what’s listed in one or two above

there so you know that gives us a nice way of

generating lots and lots of mathematical statements

based upon simple compound statements simple statements

okay so for instance if we look at this statement here this p and q or q

we have the propositional variables p and q so the propositional variables

you know you can decompose every compound statement

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uniquely into a finite number of simple statements in a unique way

and in that way we’re able to figure out what the propositional

variables are for mathematical statement and we’ll call these propositional

variables or sometimes statement variables

so we’re going to consider each logical connective in turn and say exactly what

it is but before we do that i want to get on

with the fact that i’m going to be using t’s and f’s for my tables here when i’m

defining these connectives here so some people use ones and zeros

i’m going to use t and t and f so let’s do it first up is the logical

the the negation connective and so let’s talk about the negation connective now

so the statement not p um is called the negation of the

statement p and it’s defined as the denial of the statement

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so this is going to be not p is false whenever p is true and not p is true

whenever p is false so we can put that into a nice table

and as you see right here the column the column right here is going to be

the different values that p can take on p is a propositional

variable or a mathematical statement so it can either be true or false and so

the not of p will be the opposite so in other words

the not connective converts true to false and false to true

okay so now let’s talk about the conjunction and disjunction connectives so

this is the conjunction p and q and it’s true when both p and q are true

and otherwise it’s false so we can summarize that in the table

so in this table right here we see we have

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two two propositional variables p and q so q is going to go true false true

false and p is just going to go true true false false and so that’s all the

possible combinations that we can have and so this defines the conjunction of p

and q as we said in words it’s true um if they’re both true and false otherwise

okay so conjunction has the usual meaning of end except that the two

statements need not be related so in everyday english you might use um you

know two statements and then an and in between them and there may be some

relationship between the the statements but this is not true in mathematics so

in mathematics or in propositional logic here we’re just looking at p

in terms of whether it’s true or false all on its own and we’re looking at q

whether it’s true or false all on its own and then we’re simply following this

definition here and so there’s need not be any type of relationship between p

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and q to form the and you just look at the statement

variables and look and see if they’re true or false or not so here’s an

example 1 plus 4 equals 5 1 plus 3 equals 4

and so i put an and in between them and so now i’m asking if the whole statement

right here is true or false so this one right here

is true and this right here is true so now i go and look at this table over here

we’re in this first row right here true true and true and so the and or the

conjunction is true and similarly we can say

we can say that if one plus four equals five if we call that a p

and one plus three equals four we call that a q

and then we can write it in logical symbols p and q

which is true by the first row there okay so now let’s look at the disjunction

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so the disjunction of p or q is false whenever both p and q are false

and it’s true otherwise so we can summarize that definition this

definition in a table and so we can see here that the disjunction or the or

is always true unless they’re both false okay so we can look at an example

so the disjunction is used logically in the inclusive and or sense

and you know where do we get this symbol from

so i believe it’s from the latin word vel which begins with the v

and so this symbol looks like a v in any case let’s look at an example

so we have 1 plus 4 equals five and one plus three equals four

so this one is true and this one is true and so now we just go to this column up

to this row right here true and true and true so this this is a true disjunction

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and now let’s look at this one one plus four equals five or one plus four equals

or one plus three equals five this one is false

so now we have a true and a false and that’s true and so this junk this

disjunction right here the whole statement right here with the or in it

is a true disjunction okay so now let’s look at the

implication and equivalence connection connectives so here we go

so the statement p right arrow q or it’s red p implies q

is called the implication or the conditional of p and q

and it’s false when p is true and q is false and otherwise it’s true

so you see these words up here really just

define what it is and this table makes it easier to read

when we have all the possible combinations over here true false true

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false true true false false and here’s here’s how we define the implication

if they’re both true we get true if it true implies the false is false and when

you have a false hypothesis here we end up with a true implication

so what do i mean by hypothesis so hypothesis is

the statement that comes right in front of the arrow and the q is the conclusion

and it’s the statement that comes after the arrow okay so

we can look at some examples so looking at something for example like

we have some p’s and q’s here we can go do that so here’s some p’s and q’s

[Music] so we can talk about p implies a q so this will be

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one plus two equals three implies one plus two equals four

and this is the true implies the false so this statement right here would be

false this implication right here would be false because we have a true

hypothesis and a false conclusion here so what if we looked at the implication

q implies p so this would be one plus two equals 4 implies 1 plus 2 equals 3

and then this implication right here would be

true because this is a false implies a true and as you can see the table over

here false implies a true is true okay so now we can look at something

we can look at how about if we look at not p implies the q so this would be

a negation here so not p implies q and now what would this be here so p is true

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so not p this would be false now so false implies the false that would be true

okay now what about if we took here a i’ll just do it down here what if we

took a not p implies a not q what would this implication here be

so we would have not p so not one plus two equals three implies

not one plus two equals four and so what would this statement right here be

so remember p is true so this is n the negation of a true so this would be false

and so it really wouldn’t matter what this right here is

so this would be a true statement right here okay so there’s some examples there

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there’s there’s one example and then here’s another one and here’s another one

okay so there are different ways of expressing the

conditional though so for example we have the

usual approach to everyday mathematics is to use words

so when you’re using the implication and a lot of math statements are written in

implications um you know you want to use words so so

there’s different words so we can say for the right arrow symbol we can say p

implies a q we can say if p then q those are by far the most popular way

though but sometimes we’ll abbreviate and we’ll just say if p

then q or we’ll just say if p q or q if p and then this one is probably the most

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uh challenging to understand but from a work from a perspective of

using words p only if q and then p is sufficient for q that

one’s straightforward q is necessary for p

and then the eighth way of saying this right arrow is whenever p q

and then the ninth q whenever p so people like to say things

different ways of course um so this is one way of

this is lots there’s lots of ways there’s there’s even more ways of doing

this but those by far the most common ways there okay so now let’s look at

equivalence so p left right arrow which is red p if and only if q

is called the equivalence or by conditional of p and q and and it’s true

if and only if p and q are both true or both false

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so we can put that in a table here to summarize it so as you see here

if they’re both true it’s true if they’re both false it’s true

and if it’s mixed if one is true and the other is false then it’s false

so this is the equivalent now there’s different ways of saying the

equivalence but the first thing that we should know is that we’re taught to be

equivalent means it’s when you have these two implications

so p and q are equivalent exactly when q implies p and p implies q are true

now in the next video we’re going to start

constructing truth tables and so we’ll get a better understanding of all seven

of these logical connectives this video is just introducing these

logical connectives the seven logical connectives

all right so here’s some common ways to express the biconditional

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so the first one is p if and only if q p is necessary and sufficient for q

and then p is equivalent to q and then the last one would be

p and q are equivalent so i think that those are all pretty

intuitive except perhaps number two number two is using this uh double

implication here um and some terminology that we used

when we looked at the implication so there’s four ways of also talking about

equivalence there okay so now let’s talk about the

alternative and joint connectives so here let’s summarize of what we said so far

and i’ve done this here in a truth table

so so far we’ve talked about these first four connectors right here and we just

finished talking about the fifth and so what i’ve done is i’ve summarized

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them here in a table so here we have q true false true false

and we have p true true false false and here’s the definition of the and the

definition of the or implication and equivalence

and so now i want to look at the definition of the alternative negation

and the joint negation so what are these two right here well here’s their

definition right here but i would like to give more than just

the definition i’d like to try to explain where they come from

okay so for example the alternative negation which is sometimes called the

not and or the nand so if we look at this one right here how

do we get these values right here sorry about that how do we get these values

right here so how do we get this false and true and true and true so

if i look at the and here i get true false false false

now what happens if i take the negation of the and so if i start negating all of

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these so if i negate a true i’ll get the false and if i negate all these falses

i’ll get a true so this right here the nand comes from negating the and

similarly the joint negation or the nor comes from negating the or

so here’s the or for p and q and if i go

and negate these trues we get the falses

and if i negate this false i get to true so the nor comes from negating the or

statement there now in the upcoming videos we’re going

to talk about truth tables we’re going to talk about logical equivalence

and then we’re going to talk about functionally complete

and in that video we’ll see how these are must

uh must use logical connectives logical connectives that you must understand as

well as these other five over here so i look forward to seeing you in that video

so if you have any questions or ideas please use the comment section below and

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