Cartesian Product of sets (Part - 2) - Concepts, Math, Class 11 Video Lecture

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before watching this video please watch
delicious and functions part 1 let me
now introduce to a new term called
Cartesian product of sense so what is
Cartesian product before going to the
definition let us understand the concept
let us assume there is a man there is a
set of man ramen shop and these two guys
want to marry women and they have three
choice Sita Gita and Rita now in how
many ways this guy can vary if you see
Rama can marry Sita Ram can also marry
Gita
similarly ROM can marry Rita in case of
sham sham can marry Sita charm can also
marry Gita and Shou can also marry Rita
so hi how I got us first I to gram and
then four-on-three possible combination
Rama Sita then Ram Gita and then now
Rita then second I took Shyam and then
for sham sham sheikah shemitah and champ
Rita so there are six possible ways
where man can very women in this case
when they are two men Ramana and they
want to marry three women Sita Gita and
Rita there are six possible ways in
Cartesian product also we do something
like this only then we have two sets
example set of man and set of women then
you do a Cartesian product of this what
you get is something like this
we'll discuss now the definition of
Cartesian product Cartesian product set
is the product of two so as the names of
it Cartesian product
it is nothing but product of two sets in
terms of sets the Cartesian product P
comma P into Q this is how we'll say P
into Cu is set of all ordered pairs
whose first component is member of P and
second component is member of Q so it
will be something on P by Q form the
first is from P and second is from Q so
if you see this when you write this in
the set builder form you say P into Q is
going to P Q P comma Q where P is member
of capital B and small Q is member of
capital Q from the early example we see
when man is one set women is another set
when I say man into women we got this as
the output we have seen this now let's
note some point for Cartesian product
two ordered pairs are equal if and only
if the corresponding first elements in
the second elements are equal for
example when we say there is set to
three to four this is one set and the
next set is three two three four this is
another set so both the sets are not
equal because elements are seen but the
first element matched the first element
and second element should match the
second unit that's how we compare when
we are comparing the product remember
when we accompanied the normal product
in that normal set in that case the
order does not matter but in case of
ordered Cartesian product the order does
matter the first element should match
the first element and the second element
should match to the second element when
we say a cross a cross a this is nothing
but a B C where all ABC are part of a
and this is called ordered triplet when
we say aim to finite in a and infinite
it gives a
tonight it makes sense when you are
saying a into in tonight a this is
infinite because in math also when you
say one into Infinite is infinite
similarly in sets also when you say a
into infinite set is infinite because
you'll be creating in finite number of
sets product sets also number of a cross
B number of elements in a cross B is
equal to number of elements in a into
number will be for example we have seen
in men case charm and ROM was there I
think two element in women Sita Gita and
Rita three so number of element in M
cross W where six actually which you can
see here is two into three also in the
last example we see we saw there were
three sets
ABC each one having two elements each
and the final product we caught was
having eight element so eight is equal
to a 2 into 2 into u so that means
number of element in a cross B is equal
to number of element in a into number of
element in B similarly for three items
it is number of element in a cross B
cross C is equal to number of elements
in a in a more filaments in B into
number congruence the last example we
have seen that this guy was a into
element this guy had two element C had
two element the total output we got was
having eight elements so this proves
that both are true
let me read let me reiterate in ordered
pair they are equal only if first
element is equal to the first women to
the second set and second element is
equal to the second element second check
also last point to note a cross B is not
equal to V crossing we can prove this
for example set a has one two and set B
has three so when you this is a and this
is when you say a cross B this is
nothing but 1 3 & 2 3 then you say BRE
cross a this is equal to 3 1
and three two so as we have told that
only the element does not matter the
order also matters so in this case one
and three is not equal and threes one is
not equal so a cross B is not equal to B
cross a let's solve one problem it says
X plus 1 comma y -1 this set is equal to
3 comma 1 we have to find the value of x
+ 1 so as we have seen there's two sets
are equal if first element is equal to
first segment of this first element of
second element of this set is equal to
second element of this that's why we can
say X plus 1 is equal to 3 and Y minus 2
is equal to 1 you solve this you get X
is zero Y is equal to C and this is the
a simple question just you have to note
that first element of set a is equal to
the first element of set B first and
when second element of set a is
producing an element of set B let us
take one more column where we have X and
X set X has three elements X Y Z and set
P has only one element P now we are
asked to form X cross P and P cross X so
let us form X cross P first so to make
create X cross P first let us take X
first end of X that is X so it is X and
first limit of P is only one element of
each we will take X comma piece 1 then
we will take second element of X Y comma
P and then we will think third element
of X that is Z comma so X cross P is
nothing but X comma P Y comma P and Z
comma B there are three elements there
are three different types now when we
say P cross X what we do first we take P
first P and then we say first event of X
EECOM IX then we'll take the second 11x
that is P comma Y then we will take P
third element of it is become a self so
these are the values of P Comics and you
could clearly see that X comma P is not
equal to P cross X let's say for example
we have set a is equal to one two three
three elements in set B we have two
elements three four and set C we have
three elements four five six now we are
supposed to find a cross B intersections
to do this first let me find B
intersection C what are the common
elements between V and C four four four
is the common element between B and C so
when I say B intersection C I mean set
with one element that is four and now in
sent a we have one two three so when I
say a cross B intersection C I mean to
say I say first event of a that is one
comma first element of B intersection C
that is for all this first limit the
second limit of a there is 2 comma 4 nor
third element of way that is three
former this is my answer or a cross B
intersection C now let me take one more
example when we are saying a cross B
intersection a cross C so for that first
we have to find a cross B a cross C and
then we have to do intersection of that
so let's find a cross B let me do here a
cross B will say first element of a 1
first element of B 3 correct and then we
will say first element of 1 a and second
element of B 4 first element is over now
for a never take second element 2 comma
3 second diamond of B similarly 2 comma
2nd element of B 4 now this is over so
we will take third element of a that is
3 comma first segment of B 3 and then
third
meant of a comma second element of V so
this is a cross B similarly let me find
a cross C a cross C is first element of
a 1 comma first limit of C for then
again we will keep taking the first
limit of it until illness all the
remains of C are not consumed so 1 comma
5 second element of C and 1 comma 6
third element of C now let me take
second element of a so I'll say two and
first segment of C for two second
element of C - second element of third
element of C similarly this is complete
will take third element of a so it will
be 3 comma 4 because a to C third
element of this and first statement of C
3 comma 5 and 3 comma 6 so this
completes the whole set of a cross C now
we have to find the intersection of this
that is the common element 1 comma 3 it
is is it there in this set it is not
there so we'll cut this 1 comma 4 is it
there yes it is there this is the 1
comma 4 2 comma 3 it is there it is not
there so it is 2 comma 2 comma 5 2 comma
3 is not there so well cut this to go
upward is there it is here 3 comma 3 is
there incept a cross C it is not there
we will cut this 3 comma 4 is there a
cross C yes it is there so the common
element is 1 4 2 4 & 3 4 that means we
can say a cross B intersection of a
cross C is equal to 1 comma 4 2 comma 4
and 3 comma 4 this is though
answer I hope you got the logic what we
have done we have first found a cross B
and then we have found a cross C and
then we did an intersection that is we
found the common element between these
two sets and the common element was
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