Lesson 1.1 basic ideas of sets part 2

Cardinality of Sets
- it is the number of elements in a
set.
- it is denoted by
n(A)
which is:
number of
elements
in a set
=
Set Name

Cardinality of Sets
Let’s look for the cardinality of this set
example:
J = { dog, cat, horse, cow }
n(J) = 4
Cardinality
Equal Sign
Number of Elements
We write:

“The cardinality of
Set J is 4.”
This is read as..
Cardinality of Sets

P = { triangle, circle, kite, up arrow, star }
Cardinality of Sets
n(P) =
5

What’s the Cardinality?
T = { cow, sheep}
n(T) =
H = { c, d, f, g, h, q, s, k, o, e }
n(H) =
n(K) =
2
10
4
K = {Apple, Samsung, Nokia, MyPhone }

1. 𝐹 = 𝑝𝑎𝑤𝑛, 𝑘𝑛𝑖𝑔ℎ𝑡, 𝑏𝑖𝑠ℎ𝑜𝑝, 𝑟𝑜𝑜𝑘, 𝑞𝑢𝑒𝑒𝑛, 𝑘𝑖𝑛𝑔
2. 𝐸 = {𝑎, 𝑏, 𝑐, 𝑑, 𝑒, … 𝑧}
3. 𝐵 = {𝑀𝑜𝑛𝑑𝑎𝑦, 𝑇𝑢𝑒𝑠𝑑𝑎𝑦, 𝑊𝑒𝑑𝑛𝑒𝑠𝑑𝑎𝑦, … , 𝑆𝑢𝑛𝑑𝑎𝑦}
4. 𝑅 = 𝑀𝑎𝑡ℎ, 𝐸𝑛𝑔𝑙𝑖𝑠ℎ, 𝑆𝑐𝑖𝑒𝑛𝑐𝑒, 𝐹𝑖𝑙𝑖𝑝𝑖𝑛𝑜, 𝑆𝑜𝑐𝑖𝑎𝑙 𝑆𝑡𝑢𝑑𝑖𝑒𝑠
5. 𝐴 = {𝑎, 𝑒, 𝑖, 𝑜, 𝑢}

1. 𝐹 = 𝑝𝑎𝑤𝑛, 𝑘𝑛𝑖𝑔ℎ𝑡, 𝑏𝑖𝑠ℎ𝑜𝑝, 𝑟𝑜𝑜𝑘, 𝑞𝑢𝑒𝑒𝑛, 𝑘𝑖𝑛𝑔
2. 𝐸 = {𝑎, 𝑏, 𝑐, 𝑑, 𝑒, … 𝑧}
3. 𝐵 = {𝑀𝑜𝑛𝑑𝑎𝑦, 𝑇𝑢𝑒𝑠𝑑𝑎𝑦, 𝑊𝑒𝑑𝑛𝑒𝑠𝑑𝑎𝑦, … , 𝑆𝑢𝑛𝑑𝑎𝑦}
4. 𝑅 = 𝑀𝑎𝑡ℎ, 𝐸𝑛𝑔𝑙𝑖𝑠ℎ, 𝑆𝑐𝑖𝑒𝑛𝑐𝑒, 𝐹𝑖𝑙𝑖𝑝𝑖𝑛𝑜, 𝑆𝑜𝑐𝑖𝑎𝑙 𝑆𝑡𝑢𝑑𝑖𝑒𝑠
5. 𝐴 = {𝑎, 𝑒, 𝑖, 𝑜, 𝑢}
𝑛 𝐹 = 6
𝑛 𝐸 = 26
𝑛 𝐵 = 7
𝑛 𝑅 = 5
𝑛 𝐴 = 5

Equivalent Sets
are sets containing exact
number of elements.
Then if we want to find if they are
, we are going to
compare their number of elements.
Let’s try this example..

are sets containing exact
number of elements.
Then if we want to find if they are
, we are going to
compare their number of elements.
Equivalent Sets
R = { 1, 2, 3 } S = { a, b, c }
n(R) = 3 n(S) = 3
Ex.
Therefore, Set R and Set S are
equivalent sets because they have the
or the
.
≈

“Set R is equivalent to Set” S
This is read as..
Equivalent Sets

Take note:
is
used to denote sets
are equivalent
It’s like an equal sign
but in a curvy way.. 
Equivalent Sets
Ex.
N={t,w,o} ≈ M={2,1,4}

Equivalent or Not
A = {cherry, apple, banana}
B = {pechay, ampalaya, kalabasa}

Equivalent or Not
Q = { Kanto, Unova, Hoenn }
S = { Johto, Sinnoh }

Equivalent or Not
C = {a, b, e, f, k}
D = {23, 41, 33, 67, 99}

Equivalent or Not
C = {July, August, May, February}
D = {January, June, March, April}

Equivalent or Not
E = {1, 2, 3, 4 }
F = {a, b, c, d, … }

Equal Sets
are sets with and
of .
Then again, if sets are to be considered
equal, they must have:
1. same number of elements or
cardinality
2. exactly the same elements

A = {a, b, c} B = {c, a, b}
are sets with and
of .
Then again, if sets are to be considered
equal, they must have:
1. same number of elements or
cardinality
2. exactly the same elements
Ex.
n(A) = 3 n(B) = 3
Set A has 3 elements and Set B also has 3
elements. Therefore, they have the
or . We can also
say that they are . First condition
PASSED! 
≈
To be considered as an equal sets, first they must
have the same ? Do they
have?
Equal Sets

A = {a, b, c} B = {c, a, b}Ex.
Set A contains the elements , ,and
while Set B contains the elements , ,
and . These two sets have
.
Next, these two sets must contain
.
=

Equal Sets
Therefore, we can say that..
A = { a, b, c } B = { c, a, b }
For they have the
and these
two sets contain
.

“Set A is to Set B”
Once again, this is read as..
Equal Sets

Take note:
is used
to denote they sets
are equal.
Equal Sets
Ex.
N={a, p, l, e} = M={e, l, p, a}

Equal or Not Equal
A = {beans, peanuts, kalabasa}
B = {peanuts, beans, kalabasa}

Equal or Not Equal
C = {July, August, May, February}
D = {January, June, March, April}

Equal or Not Equal
A = { 1, 2, 3, 4, 5 }
B = { 5, 4, 3, 2, 1… }

Equal or Not Equal
A = { Chito M., Ely B., Rico B.}
B = { Rico J., Chito M., Ely B.}

Equal or Not Equal
A = { Buruguduyistunstuguydunstuy }
D = { Buruguduystunstuguydunstuy }

This is the symbol.
Subsets
Example:
Read as.. “ is a of ”

There are cases where two or more sets
contain some, but not all of the same
elements.
Consider the :
A = { 2, 4, 6, 8, … }
and the
B = { 1, 2, 3, 4, … }
Subsets
⊆ = Subset

We can say that:
2 ∈ A and 2 ∈ B
8 ∈ A and 8 ∈ B
Consider the :
A = { 2, 4, 6, 8, … }
and the
B = { 1, 2, 3, 4, … }
There are cases where two or more sets,
contain some, but not all of the same
elements.
Subsets
⊆ = Subset

In fact, every element that is in A is also
contained in Set B.
Therefore, we can say that is
, or in symbols, we can write
is a of
Subsets
⊆ = Subset

Again, is a of , if
and only if,
.
A = { 2, 4, 6, 8, … } B = { 1, 2, 3, 4, … }
In fact, every element that is in A is also
contained in Set B.
Therefore, we can say that is
, or in symbols, we can write
is a of
Subsets
⊆ = Subset

to be
considered a .
J = { a, e, i, o, u } F = { a, b, c, ... z }
∴ J ⊆ F
For this example let’s use Set J and Set F:
Subsets
⊆ = Subset

K = { 2, 1, 10, 14 } L = { 1, 2, 3, 4 ... 9 }
∴ K ⊆ L
Another example:
to be
considered a .
Subsets
⊆ = Subset

Take note also that
C = { 1, 2, 3 }
Example:
C = { 1, 2, 3 }
∴ C ⊆ C
Subsets
⊆ = Subset

Take note also that
Example:
C = { 1, 2, 3 }D = { }
∴ D ⊆ C
Subsets
⊆ = Subset

This is the symbol.
Proper Subsets
Example:
Read as.. “ is a of ”
⊆ = Subset
⊂ = Proper Subset

If is a of
, written as
Then it must satisfy this two conditions:
* must be a subset
* must contain
in
Proper Subsets
⊆ = Subset
⊂ = Proper Subset

If is a of
, written as
Then it must satisfy this two conditions:
* must be a subset
* must contain
in
Example:
B = { a, b, c, d}A = { a, b, c }
∴
Proper Subsets
⊆ = Subset
⊂ = Proper Subset

D = { rat, cat, cow }C = { rat }
∴
Now, let’s use Set C and Set D:
Proper Subsets
⊆ = Subset
⊂ = Proper Subset

to be considered a .
* must be a subset
* must contain in
Read first and tell whether each
statement is TRUE or FALSE.
{a,b,c} ⊆ {a,b,c,}
⊆ = Subset
⊂ = Proper Subset

* must be a subset
* must contain in
{a,b,c} ⊂ {a,b,c,}

* must be a subset
* must contain in
{10, 30} ⊆ {1, 2, 3, … 100}

* must be a subset
* must contain in
{∅} ⊆ { }

* must be a subset
* must contain in
{ 102 } ⊆ { 2, 4, 6, 8, … }

* must be a subset
* must contain in
{4,2,1} ⊂ {2, 1, 3, 4}

Determining
the Possible
Subsets of a
Set

Determining the Possible
Subsets of a set
Let’s take for example this set. { 1, 2 }
Solution:
2. We also know that every set is
a subset of itself.
Possible Subsets
{ }1. We know that the empty set is
the subset of all sets.
{ 1, 2 }
3. Form all the subsets with 1
element, with 2, with 3 and so on
and so forth depending on the
number of elements.
{ 1 }
{ 2 }
Number of subsets: 4

List All The Possible Subsets
And Write The Number Of
Total Subsets.
{ apple, banana, mango}
{ }
{apple, banana,
mango}
{apple} {banana} {mango}
{apple , banana} {apple, mango} {banana, mango}
Number of subsets: 8
I.
II.
III.

Determining the Possible
Subsets of a set
Let’s try to find the possible
subsets of this one..
{ m, a, t, h }

Universal Set
The , denoted by U, is
the to
any set used in the problem.
A={a,b,c,d}
C={f,h,i,j}
B={c,d,e,f}
U={a,b,c,d,e,f,h,i,j}

Universal Set
The universal set can change from
problem to problem, depending on the
nature of the set being discussed.
A={1,2,3,4}
C={5,3}
B={6,7,8}
U={1,2,3,4,5,6,7,8,9}

Complement of a Set
The of a set A, written as
is the set of all the elements in the
that

U
Given:
A
Find A’
is the set of elements in
U but not in A. Let’s try this example.

Complement of a Set
A’ =
“The of A are
the elements
and .
This is read as..

How Do We Write
Complement?
U = { ears, eyes, nose, lips, skin, cheeks}
A = { eyes, nose, cheeks}
A = { ears, lips, skin}
Set Name Equal Sign
Opening Brace
Element/s Closing Brace
‘Apostrophe
You write:

U = { 1, 2, 3, 4, 5 }, A = { 1, 3, 5 }
Given:
∴ A’= { 2, 4 }
Find:
Complement of a Set
A’ =
Example:

U = { 1, 2, 3, 4, 5 }, B = { 1, 5 }
Given:
∴ B’= { 2, 3, 4 }
Find:
Complement of a Set
Let’s try another example:
B’ =

U = { 1, 2, 3, 4, 5 }, C = { }
Given:
∴ C’= { 1, 2, 3, 4, 5 } = U
Find:
Complement of a Set
Another one.. 
C’ =

Complement of Set
U = { adobo, sinigang, menudo, afritada,
tinola, lechon }
A = { sinigang, tinola, adobo }
A’ = { menudo, afritada, lechon }

Complement of Set
U = { Tepig, Victinni, Oshawott,
Reshiram, Snyvil, Zekrom }
A = { Victinni, Reshiram, Zekrom}
A’ = { Tepig, Oshawott, Snyvil}

Complement of Set
U = { n, m, a, w, r, p, u, e, t, o}
A = {m, e, p, w}
A’ = { n, a, r, u, t, o}

Lesson 1.1 basic ideas of sets part 2

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Lesson 1.1 basic ideas of sets part 2