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Typology: Summaries
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โ RULE #1:
class rules
โ RULE #3:
โ RULE #2: โ RULE #4:
learning objectives
notation;
SET It is a collection of related and well-defined objects
called elements (denoted by โ).
Georg Cantor (1845-1918) introduced the word set in 1879.
Historical background
๐ด = { 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 }
๐ต = {๐๐๐, ๐ฆ๐๐๐๐๐ค, ๐๐๐ข๐}
๐ถ = {๐ฅ| 0 < ๐ฅ < 5 }
๐ถ = { 1 , 2 , 3 , 4 }
WRITING A SET
is a way of listing the elements
separated by a comma.
Example:
Let ๐ด and ๐ต be sets. If ๐ด is the set of all even whole numbers
between 1 and 10 , and ๐ต is the set of all odd whole
numbers between 1 and 10 write set ๐ด and ๐ต in roster
notation.
Solution:
WRITING A SET
it is a way of representing or explaining
the properties that must satisfy by the
elements of a set.
Example:
Consider ๐ and ๐ as sets of natural numbers.
Let ๐ = { 1 , 2 , 4 , 3 , 5 , 6 , 7 } and ๐ = 11 , 12 , 13 , 14.
Write the sets in set-builder notation.
Solution:
EQUAL SETS
Equal sets are sets whose elements are exactly
the same, otherwise, the sets are unequal.
EQUIVALENT SETS
Examples:
Equivalent sets are sets having the same cardinal number,
otherwise they are non-equivalent sets.
Given the following sets:
P = {x/x is a letter from the word โtasteโ}.
R = {x/x is a letter from the word โeatsโ}.
S = {x/x is a letter from the word โtestโ}.
Which two sets are equal? Which sets are equivalent?
SUBSET
These are sets contained in a universal set
or another set.
Definition.
If ๐ด and ๐ต are sets, then ๐ด is called a subset of ๐ต, denoted by
๐ด โ ๐ต, if and only if, every element of ๐ด is also an element of
๐ต. Symbolically,
๐ด โ ๐ต means that for all elements ๐ฅ โ ๐ด, then x โ ๐ต.
๐ด โ ๐ต is read as โ๐ดโ is a subset of โ๐ตโ.
๐ด โ ๐ต is read as โ๐ดโ is not a subset of โ๐ตโ.
SUBSET
These are sets contained in a universal set
or another set.
Solution:
Find the subsets of the following sets.
a. P = ๐ , ๐ข, ๐
b. Q = ๐ , ๐, ๐, ๐
c. S = ๐, ๐, ๐, ๐, ๐ก
, ๐ , ๐ข , ๐ , ๐ , ๐ข , ๐ , ๐ , ๐ข, ๐ , {๐ , ๐ข, ๐}
, ๐ , ๐ , ๐ , ๐ , ๐ , ๐ , ๐ , ๐ , ๐ , ๐ , ๐, ๐ ,
๐, ๐ , ๐, ๐ , ๐ , ๐, ๐ , ๐ , ๐, ๐ , ๐ , ๐, ๐ , ๐, ๐, ๐ ,
{๐ , ๐, ๐, ๐}
How many subsets can be obtained
here without enumeration?
How many subsets are there if the sets has:
1 element
2 elements
3 elements
4 elements
= 2 subsets
= 4 subsets
= 8 subsets
= 16 subsets
Thus, to get the number
of subsets of a given set,
use the formula ๐
๐ .
For example, if ๐ ๐จ = ๐
elements,
then ๐
๐ = ๐๐ subsets.
NUMBER OF SUBSETS
POWER OF A SET
It is the set of all subsets for any given
set which includes the empty set.
Example:
Consider ๐ = {๐, ๐, ๐๏ฝ. Let ๐(๐) denotes
the power of a set ๐. So,
Based on ๐ and ๐(๐) tell whether each of
the following is TRUE or FALSE.
P ( S ) =๏ป ๏ฆ,๏ป ๏ฝ ๏ป ๏ฝ ๏ป ๏ฝ ๏ป a , r , i , a , i ๏ฝ ๏ป, a , r ๏ฝ ๏ป, i , r ๏ฝ ๏ป, i , r , a ๏ฝ๏ฝ
a ๏ P ( S ) ๏ป ๏ฝ a^ ๏^ P ( S )
๏ป๏ป ๏ฝ a ๏ฝ ๏ P ( S ) ๏ป a , r ๏ฝ ๏ P ( S )
Determine whether the statement is true or false.
UNION OF SETS
The union of two or more sets contains
ALL the elements in all the sets under
consideration.
Suppose ๐ด and ๐ต are sets. The union
of sets ๐ด and ๐ต is denoted by ๐จ โช ๐ฉ.
Example:
Consider ๐ด = 1 , 2 , 3 , 4 , 5 , 6 and ๐ต = 2 , 4 , 6 , 8 , 10 , 12.
Find ๐ด โช ๐ต.
INTERSECTION OF SETS
The intersection of two or more sets
contains the common elements in all the
sets under consideration.
Suppose ๐ด and ๐ต are sets. The intersection
of sets ๐ด and ๐ต is denoted by ๐จ โฉ ๐ฉ.
Example:
Consider ๐ด = 1 , 2 , 3 , 4 , 5 , 6 and ๐ต = 2 , 4 , 6 , 8 , 10 , 12.
Find ๐ด โฉ ๐ต.
