### Video Transcript

π is the set of all integers from
one to π. πΈ is the set of even numbers. π is the set of multiples of
five. π is the set of prime numbers. Part a, explain why there is only
one number in πΈ intersection π. Part b, given that π is a multiple
of 10, how many numbers are in π intersection π? Circle your answer. Is it π divided by five minus one,
π divided by five plus one, π minus five, or π divided by five?

Letβs just remind ourselves of some
of the notation that we need to know. This is the Greek letter π. And itβs usually used to denote the
universal set. Thatβs all the objects weβre
interested in. And here, those are all
integers. Remember, those are whole numbers
from one to π. So we could say that that could be
one, two, three, four, all the way through to π.

We then see that πΈ is the set of
all even numbers. Remember, an even number is a
multiple of two. And it will always end in two,
four, six, eight, or zero. π is the set of multiples of
five. And those are the numbers in the
five times tables. Theyβre five, 10, 15, 20, and so
on. And then, we have π as being the
set of prime numbers. Remember, a prime number has
exactly two factors, one and itself. The first five prime numbers are
two, three, five, seven, and 11. Itβs important to remember that one
is not a prime number. One actually only has one factor,
one. So how does this help us for part
a?

Well, this symbol here that looks a
little bit like the letter n means the intersection of two sets. When we think about that in terms
of two circles on a Venn diagram, itβs the overlap. So weβre interested in the numbers
that would appear in the overlap between the set of even numbers and the set of
prime numbers. That is the set of numbers that are
both even numbers and prime numbers. Two is the only even prime
number. This means it is the only number
which will appear in the intersection between sets πΈ and π.

Remember, we said a prime number
has exactly two factors. Two has two factors; itβs one and
two. Any other even number will have a
factor of two apart from one and itself. So thereβs no way that any other
even number can have only two factors. And we can now see that there can
only be one number in πΈ intersection π. Itβs the number two, since two is
the only even prime number.

Now, letβs consider part b. Weβre told that π is a multiple of
10. Remember, multiples of 10 are the
numbers in the 10 times table: 10, 20, 30, and so on. And weβre looking to find how many
numbers there are in the intersection of sets π and π. That means weβre interested in the
multiples of five which are also integers from one to π. If π is equal to 10, then numbers
that appear in this set are five and 10. If π is equal to 20, the multiples
of five in this set are five, 10, 15, and 20. And if π is equal to 30, the
multiples are five, 10, 15, 20, 25, and 30.

That means that when π is equal to
10, there are two numbers in the set. When π is equal to 20, there are
four numbers in the set. And when π is equal to 30, there
are six numbers in the set. In fact, we can see that for every
10 numbers there are in π, there are two multiples of five. When π is a multiple of 10, then
we can say that two out of every 10 of these numbers are multiples of five. Thatβs two-tenths of π.

Remember, βofβ is commonly
interchanged with the multiplication symbol. And we can simplify two-tenths to
one-fifth. And we say that one-fifth of π are
multiples of five when π is a multiple of 10. Another way of saying this is π
over five. So our answer here is π over
five.