7 Logical Connectives That You MUST Master

Video Series: Logic and Mathematical Proof (In-Depth Tutorials for Beginners)

(D4M) — Here is the video transcript for this video.

00:00
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

00:01
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

00:02
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

00:03
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

00:04
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

00:05
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

00:06
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

00:07
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

00:08
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

00:09
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

00:10
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

00:11
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

00:12
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

00:13
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

00:14
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

00:15
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

00:16
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

00:17
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

00:18
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

00:19
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

00:20
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

00:21
don’t forget to check out the series logic and mathematical proof in-depth
tutorials for beginners and i want to say thank you for watching and i’ll see
you next time if you like this video please press this
button and subscribe to my channel now i want to turn it over to you math can be
difficult because it requires time and energy to become skills i want you to
tell everyone what you do to succeed in your studies either way let us know what
you think in the comments

About The Author
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David A. Smith (Dave)

Mathematics Educator

David A. Smith is the CEO and founder of Dave4Math. His background is in mathematics (B.S. & M.S. in Mathematics), computer science, and undergraduate teaching (15+ years). With extensive experience in higher education and a passion for learning, his professional and academic careers revolve around advancing knowledge for himself and others. His work helps others learn about subjects that can help them in their personal and professional lives.

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