All Set Theory Resources
Example Questions
Example Question #1 : Axiomatic Set Theory
Let denote all straight lines in the Cartesian plane. Does
,
, or both belong to
?
is a set that contains all straight lines that live in the Cartesian plane, this is a vast set. To determine if
,
, or both belong to
, identify if the elements of each set create a straight line, and if so, then that set will be a subset of
. In other words, the set will belong to
.
Identify the elements in first.
This statement reads, contains the
coordinate pairs that live on the line
.
Since
is a straight line that lives in the Cartesian plane, that means
belongs to
.
Now identify the elements in .
This means that the elements of are 2, 4, 6, and 9. These are four, individual, values that belong to
. They do not create a line in the Cartesian plan and thus
does not belong to
.
Therefore, answering the question, belongs to
.
Example Question #2 : Axiomatic Set Theory
Let denote all parabolas in the Cartesian plane. Does
,
, or both belong to
?
is a set that contains all parabolas that live in the Cartesian plane, this is a vast set. To determine if
,
, or both belong to
, identify if the elements of each set create a straight line, and if so, then that set will be a subset of
. In other words, the set will belong to
.
Identify the elements in first.
This statement reads, contains the
coordinate pairs that live on the parabola
.
Since
is a parabola that lives in the Cartesian plane, that means
belongs to
.
Now identify the elements in .
This means that the elements of are those that live on the straight line
. Thus
does not create a parabola in the Cartesian plan therefore
does not belong to
.
Therefore, answering the question, belongs to
.
Example Question #1 : Set Theory
Determine if the following statement is true or false:
In accordance to primitive concepts and notations in set theory, many axioms lead to paradoxes.
False
True
False
First recall the primitive concepts and notations for set theory.
"class", "set", "belongs to"
Now, when deciding what constitutes a primitive concept, it is agreed upon in the math world that four main criteria must be met.
1. Undefined terms and axioms should be few.
2. Axioms should NOT be logically deducible from one another unless clearly expressed.
3. Axioms are able to be proved.
4. Axioms must NOT lead to paradoxes.
Thus, the statement in question is false by criteria four.
Example Question #2 : Set Theory
Determine if the following statement is true or false:
In accordance with primitive concepts and notations in set theory, many axioms are deducible from other axioms.
True
False
False
First recall the primitive concepts and notations for set theory.
"class", "set", "belongs to"
Now, when deciding what constitutes a primitive concept, it is agreed upon in the math world that four main criteria must be met:
1. Undefined terms and axioms should be few.
2. Axioms should NOT be logically deducible from one another unless clearly expressed.
3. Axioms are able to be proved.
4. Axioms must NOT lead to paradoxes.
Thus, the statement in question is false by criteria two.
Example Question #3 : Set Theory
Which of the following describes the relationship between the inhabited sets and
if
?
and
have equal cardinality.
and
intersect.
and
are disjoint.
is a subset of
.
is a subset of
.
and
are disjoint.
If the intersection of two sets is equal to the empty set (they do not intersect, i.e. they share no elements), then the two sets are said to be disjoint.
Example Question #3 : Axiomatic Set Theory
Which of the following represents , where
,
, and
?
To solve this problem, we first find the union of and
; this is the set of all elements in both sets, or
.
is simply the set of all even natural numbers. The intersection of these two sets is therefore the set of the even numbers present in
, which is the set containing the numbers 2, 8, and 10.
Example Question #3 : Set Theory
Which of the following represents , where
,
, and
?
Because and
share no elements, their intersection is
, such that
The union of
and any set is the set itself. Therefore,
.
Example Question #4 : Set Theory
For two sets, and
, which of the following correctly expresses
?
The sum of the cardinalities of two sets is equal to the sum of the cardinalities of their intersection and union. For instance, if and
:
and,
Example Question #1 : Relations, Functions And Cartesian Product
Determine if the following statement is true or false:
If and
then
.
True
False
True
Assuming ,
,
, and
are classes where
and
.
Then by definition,
the product of and
results in the ordered pair
where
is an element is the set
and
is an element in the set
or in mathematical terms,
and likewise
Now,
therefore,
.
Thus by definition, this statement is true.
Example Question #2 : Relations, Functions And Cartesian Product
Determine if the following statement is true or false:
If be the set defined as,
then
.
False
True
False
Given is the set defined as,
to state that , every element in
must contain
.
Looking at the elements in is is seen that the first two elements in fact do contain
however, the third element in the set,
does not contain
therefore
.
Therefore, the answer is False.
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