71 basic languages of sets

The Basic Language of Sets
A set in mathematics is a collection of items (real or abstract).

A set may be given by a description or we may list the items of
the set in a “{ }”.

For example, we may define the set M to be
“the coins in Maria’s purse”,

“the coins in Maria’s purse”, and if Maria’s purse contains
exactly a penny, a dime and a quarter, then we may list M as
{p, d, q}.

{p, d, q}. Each item in the set M is called an element of M.

If x is an element of the set M, we write that x ϵ M

If x is an element of the set M, we write that x ϵ M and similarly
y ϵ M means that y is not an element in M.

Hence p ϵ M but King Kong ϵ M.

An element may be listed once only.

An element may be listed once only. If Maria has two quarters in
the purse, then we must list M as {p, d, q1, q2}.

If we are to list the set
T = {different types of coins in Maria’s purse},

If x is an element of the set M, we write that x ϵ M.
Similarly y ϵ M means that y is not an element in M.
If we are to list the set
T = {different types of coins in Maria’s purse},
then T = {P, Q, D} where P, Q, and D represent the types of the
coins regardless M = {p, d, q} or {p, d, q1, q2}.

The number of elements in the set A is called the order of A
and it’s denoted as o(A).

If M = {p, d, q}, then o(M) = 3.

If M = {p, d, q}, then o(M) = 3.
If M = {p, d, q1, q2}, then o(M) = 4.

If M = {p, d, q}, then o(M) = 3.
If M = {p, d, q1, q2}, then o(M) = 4.
The empty set { } which contains nothing is denoted as Φ
and that o(Φ) = 0.

If M = {p, d, q}, then o(M) = 3.
If M = {p, d, q1, q2}, then o(M) = 4.
and that o(Φ) = 0.
A set A is said to be finite if o(A) is finite, i.e. we can finish
counting all the items in A.

If M = {p, d, q}, then o(M) = 3.
If M = {p, d, q1, q2}, then o(M) = 4.
and that o(Φ) = 0.
counting all the items in A. A set that is not finite is said to be
infinite.

If M = {p, d, q}, then o(M) = 3.
If M = {p, d, q1, q2}, then o(M) = 4.
and that o(Φ) = 0.
infinite. If B is infinite, we write that o(B) = ∞.

If M = {p, d, q}, then o(M) = 3.
If M = {p, d, q1, q2}, then o(M) = 4.
and that o(Φ) = 0.
infinite. If B is infinite, we write that o(B) = ∞. The set of natural
numbers N = {1, 2, 3,…} is not finite.

If M = {p, d, q}, then o(M) = 3.
If M = {p, d, q1, q2}, then o(M) = 4.
and that o(Φ) = 0.
numbers N = {1, 2, 3,…} is not finite. We can not finish listing
them.

If M = {p, d, q}, then o(M) = 3.
If M = {p, d, q1, q2}, then o(M) = 4.
and that o(Φ) = 0.
them. Every time we list one number, there is the next one
inline to be listed.

If M = {p, d, q}, then o(M) = 3.
If M = {p, d, q1, q2}, then o(M) = 4.
and that o(Φ) = 0.
inline to be listed. Hence o(N) = ∞.

If M = {p, d, q}, then o(M) = 3.
If M = {p, d, q1, q2}, then o(M) = 4.
and that o(Φ) = 0.
Sets related to geometry are often defined by inequalities and
drawn with pictures.

If M = {p, d, q}, then o(M) = 3.
If M = {p, d, q1, q2}, then o(M) = 4.
and that o(Φ) = 0.
drawn with pictures. For example,
if K = {x | 0 < x < 2} or {x : 0 < x < 2},
“ such that”

If M = {p, d, q}, then o(M) = 3.
If M = {p, d, q1, q2}, then o(M) = 4.
The empty set { } which contains nothing is denoted as Φ;
therefore o(Φ) = 0.
drawn with pictures. For example, 0 x 2
if K = {x | 0 < x < 2} or {x : 0 < x < 2}, The set K
then o(K) is also ∞ and K can be drawn as shown.

Example A. Farmer Andy’s family

Given a photo of Farmer Andy’s
family, answer the following
questions. Andy Beth Cathy Dan
a. Let U be the set of Farmer Andy’s
family members. List U and find o(U).


Using the 1st letter of their names, we may express Farmer
Andy’s family as U = {a, b, c, d} and that o(U) = 4.


b. Let R be the set of members that are wearing something
red. List R find o(R).


R = {a, b, d} and that o(R) = 3.


c. Let N = {non–adult(s) that is wearing something green}.
List N and find o(N).


N = {c} and that o(N) = 1.


d. List T = {members that are not wearing shoes} and find o(T).


d. List T = {members that are not wearing shoes} and find o(T).
Everyone is wearing shoes, so T = Φ = { } and that o(T) = 0.

In mathematics, the set U that Farmer Andy’s family

holds all the elements is called
the universal set.
Andy Beth Cathy Dan


the universal set.
The universal set U is the same Andy Beth Cathy Dan
as “the population” in Statistics which describe the entire set
of objects under the study.


the universal set.
of objects under the study. Hence if Farmer Andy’s family is
our sole interest, then the universal set U, or the population,
is {a, b, c, d}.


the universal set.
is {a, b, c, d}. U
Given two sets A and U, we say that A
A is a subset of U, and we write that
A B, if every element in A is also in U.
∩

A U

∩


the universal set.
is {a, b, c, d}. U
A B, if every element in A is also in U.
∩

If A U but A ≠ B, i.e. U contains element(s) A U
∩

∩
besides the ones in A, then we say that A is a proper subset
of B and we write that A U.
∩


the universal set.
is {a, b, c, d}. U
A U, if every element in A is also in U.
∩

If A U but A ≠ B, i.e. U contains element(s) A U
∩

∩
besides the ones in A, then we say that A is a proper subset
of B and we write that A U.
∩

Given two finite sets A and B,
if A B then o(A) ≤ o(B), if A B then o(A) < o(B).
∩

∩

Hence R = {a, b, d} Farmer Andy’s family

is a proper subset of
U = {a, b, c, d}, or R U.

∩
Andy Beth Cathy Dan


U = {a, b, c, d}, or R U.

∩ ∩
Also the set N = {c} U. Andy Beth Cathy Dan


U = {a, b, c, d}, or R U.

∩ ∩
Note that “{c} U” and “c ϵ U” have different meanings.
∩
and that “{c} ϵ U” and “c U” do not make sense.

∩


U = {a, b, c, d}, or R U.

∩ ∩
∩

∩
It we wish to study the fast food habit of Americans,
then the population of the study is U = {all American}.


U = {a, b, c, d}, or R U.

∩ ∩
∩

∩
Subsets of U extracted with specific attributes are called
sample sets or samples.


U = {a, b, c, d}, or R U.

∩ ∩
∩

∩
sample sets or samples. For example the set
A = {all adult American} is a sample set from U.


U = {a, b, c, d}, or R U.

∩ ∩
∩

∩
A = {all adult American} is a sample set from U. Sample sets
may also be selected randomly.


U = {a, b, c, d}, or R U.

∩ ∩
∩

∩
may also be selected randomly. A random sample S of size 25
would be a subset of 25 elements selected randomly from U.


U = {a, b, c, d}, or R U.

∩ ∩
∩

∩
may also be selected randomly. A random sample S of size 25
Example B. List all the possible sample sets of size 2 from
Farmers Fred’s family U.


U = {a, b, c, d}, or R U.

∩ ∩
∩

∩
then the population of the study is U = {all Americans}.
A = {all adult Americans} is a sample set from U. Sample sets
may also be selected randomly. A “random sample S of size 25”
Example B. List all the possible sample sets of size 2 from
Farmers Fred’s family U.
There are six of them {a, b}, {a, c}, {a, d}, {b, c}, {b, d}, {c, d}.

Sets come in 2 types; discrete and continuous.

Recall that the set M = {coins in Maria’s purse} may be listed
explicitly as {p, d, q}.

explicitly as {p, d, q}. A finite set may always be listed
completely as {x1, x2, .., xn}.

completely as {x1, x2, .., xn}. The set of natural numbers
N = {1, 2, 3,…} may also be listed in the sense that every
natural number appears in this list.
We use the “…” to
mean “to continue in
the same manner”.

natural number appears in this list. No matter how large a
number is, say 1,0001,000,000,

number is, say 1,0001,000,000, if we start to list N orderly as
{1, 2, 3,... }, we know that 1,0001,000,000 would appear down the
line.

line. These listable sets are said to be discrete.

But the set K = {x | 0 < x < 2} 0 x
2
which contains fractions ½, ¾ … and The set K
irrational numbers such as 0.123401..,√2,
is an infinite set that is packed so tightly that there is no orderly
manner to list all these numbers–as we did for M or N.

But the set K = {x | 0 < x < 2}, 0 x
2
which contains fractions ½, ¾ … and The set K
irrational numbers such as 0.123401..,√2,
is an infinite set that is packed so tightly that there is no orderly
manner to list all these numbers–as we did for M or N.
A set such as K whose elements may be packed into intervals
are said to be continuous.

Similarly in Statistics, sample sets are classified as discrete or
continuous accordingly.

Example C. a. Farmer Andy is interested in studying the eggs n
that he found daily, is this type of data discrete or continuous?

Since the number of eggs n could be {0, 1, 2,... } from a
discrete set, hence the sample data is discrete

b. Farmer Andy is interested in studying the health of the apple
trees by randomly selecting an apple daily and measure its
weight w, is this type of data discrete or continuous?

This type of data is continuous because w could be anywhere
between 0 up to, say, 2 lb, i.e. w ϵ K = {w | 0 < w < 2}.

The elements in a discrete set may be line up and counted like
sheep.

The elements in a discrete set may be line up and counted like
sheep. The elements in a continuous set I are packed so tightly
into an interval(s) that there is no way to separate and line up all
the numbers in I. Physical measurements such as length and
weight which may be measured to any precision are continuous.

71 basic languages of sets

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