Relations, Functions and Cartesian Product - Set Theory
Card 1 of 40
Determine if the following statement is true or false:
If 
and 
then 
.
Determine if the following statement is true or false:
If
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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.
Assuming
Then by definition,
the product of
and likewise
Now,
therefore,
Thus by definition, this statement is true.
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Determine if the following statement is true or false:
If 
be the set defined as,
then
.
Determine if the following statement is true or false:
If
then
Tap to reveal answer
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.
Given
to state that
Looking at the elements in
Therefore, the answer is False.
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Determine if the following statement is true or false:
If be the set defined as,
then .
Determine if the following statement is true or false:
If
then
Tap to reveal answer
Given is the set defined as,
to state that , every element in must contain .
Looking at the elements in is is seen that all three elements in fact do contain therefore 
.
Thus, the answer is True.
Given
to state that
Looking at the elements in
Thus, the answer is True.
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For a bijective function 
from set 
to set 
defined by 
, which of the following does NOT need to be true?
For a bijective function
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For a bijective function, every element in set 
must map to exactly one element of set 
, so that every element in set 
has exactly one corresponding element in set 
. All of the conditions presented must be true in order to satisfy this definition.
For a bijective function, every element in set
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For an injective function 
from set 
to set 
defined by 
, which of the following does NOT need to be true?
For an injective function
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For an injective function, every element of 
must map to exactly one element of 
. Additionally, every element in 
must map to a different element in 
so that no element in 
has multiple pairings to elements in 
. It is not necessary for all elements of 
to be connected to an element in 
.
For an injective function, every element of
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For a surjective function from set to set defined by , which of the following does NOT need to be true?
For a surjective function
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For a surjective function, every element of 
must map to exactly one element of 
. Additionally, all elements of 
to be paired with an element in 
, even if one or more elements of 
is connected to multiple elements in 
.
For a surjective function, every element of
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What type of function is 
where 
?
What type of function is
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Because multiple elements of 
can map to a single element of 
(e.g. -2 and 2 map to 2), this function is surjective.
Because multiple elements of
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What type of function is 
where 
?
What type of function is
Tap to reveal answer
Because every element of 
maps to a single element of 
, but there are many elements of 
that do not pair with any element of 
, this function is injective.
Because every element of
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What type of function is 
where 
?
What type of function is
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Because this operation is not defined for the negative elements of 
, it is either considered to be a partial function or not a function at all! Each element in the sub-domain of 
for which the operation is defined maps to exactly 1 element in 
; however, many elements of 
are not paired to any element in this sub-domain of 
, thus the function could be considered a partial injective function.
Because this operation is not defined for the negative elements of
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For which of the following pairs is the cardinality of the two sets equal?
For which of the following pairs is the cardinality of the two sets equal?
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The cardinality of 
(
) is greater than that of 
(
,) as established by Cantor's first uncountability proof, which demonstrates that 
. The cardinality of the empty set is 0, while the cardinality of 
is 1. 
, while 
. For sets 
and 
, where there exists an injective, non-surjective function 
, 
must have more elements than 
, otherwise the function would be bijective (also called injective-surjective). Finally, for 
, the cardinality of both sets is equal to the cardinality of 
.
The cardinality of
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Determine if the following statement is true or false:
If and then .
Determine if the following statement is true or false:
If
Tap to reveal answer
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.
Assuming
Then by definition,
the product of
and likewise
Now,
therefore,
Thus by definition, this statement is true.
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Determine if the following statement is true or false:
If be the set defined as,
then
.
Determine if the following statement is true or false:
If
then
Tap to reveal answer
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.
Given
to state that
Looking at the elements in
Therefore, the answer is False.
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Determine if the following statement is true or false:
If be the set defined as,
then .
Determine if the following statement is true or false:
If
then
Tap to reveal answer
Given is the set defined as,
to state that , every element in must contain .
Looking at the elements in is is seen that all three elements in fact do contain therefore .
Thus, the answer is True.
Given
to state that
Looking at the elements in
Thus, the answer is True.
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For a bijective function from set to set defined by , which of the following does NOT need to be true?
For a bijective function
Tap to reveal answer
For a bijective function, every element in set must map to exactly one element of set , so that every element in set has exactly one corresponding element in set . All of the conditions presented must be true in order to satisfy this definition.
For a bijective function, every element in set
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For an injective function from set to set defined by , which of the following does NOT need to be true?
For an injective function
Tap to reveal answer
For an injective function, every element of must map to exactly one element of . Additionally, every element in must map to a different element in so that no element in has multiple pairings to elements in . It is not necessary for all elements of to be connected to an element in .
For an injective function, every element of
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For a surjective function from set to set defined by , which of the following does NOT need to be true?
For a surjective function
Tap to reveal answer
For a surjective function, every element of must map to exactly one element of . Additionally, all elements of to be paired with an element in , even if one or more elements of is connected to multiple elements in .
For a surjective function, every element of
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What type of function is where ?
What type of function is
Tap to reveal answer
Because multiple elements of can map to a single element of (e.g. -2 and 2 map to 2), this function is surjective.
Because multiple elements of
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What type of function is where ?
What type of function is
Tap to reveal answer
Because every element of maps to a single element of , but there are many elements of that do not pair with any element of , this function is injective.
Because every element of
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What type of function is where ?
What type of function is
Tap to reveal answer
Because this operation is not defined for the negative elements of , it is either considered to be a partial function or not a function at all! Each element in the sub-domain of for which the operation is defined maps to exactly 1 element in ; however, many elements of are not paired to any element in this sub-domain of , thus the function could be considered a partial injective function.
Because this operation is not defined for the negative elements of
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For which of the following pairs is the cardinality of the two sets equal?
For which of the following pairs is the cardinality of the two sets equal?
Tap to reveal answer
The cardinality of () is greater than that of (,) as established by Cantor's first uncountability proof, which demonstrates that . The cardinality of the empty set is 0, while the cardinality of is 1. , while . For sets and , where there exists an injective, non-surjective function , must have more elements than , otherwise the function would be bijective (also called injective-surjective). Finally, for , the cardinality of both sets is equal to the cardinality of .
The cardinality of
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