# Set Theory : Relations, Functions and Cartesian Product

## Example Questions

### Example Question #7 : Set Theory

Determine if the following statement is true or false:

If  and  then .

True

False

True

Explanation:

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 #8 : Set Theory

Determine if the following statement is true or false:

If  be the set defined as,

then

.

True

False

False

Explanation:

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 .

### Example Question #1 : Relations, Functions And Cartesian Product

Determine if the following statement is true or false:

If be the set defined as,

then .

True

False

True

Explanation:

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 .

### Example Question #2 : Relations, Functions And Cartesian Product

For a bijective function  from set  to set  defined by , which of the following does NOT need to be true?

Every element of  must map to one or more elements of .

All of the conditions here must be true.

Every element of  must map to one or more elements of .

No element of  may map to multiple elements of .

No element of  may map to multiple elements of .

All of the conditions here must be true.

Explanation:

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.

### Example Question #1 : Relations, Functions And Cartesian Product

For an injective function  from set  to set  defined by , which of the following does NOT need to be true?

No element of  may map to multiple elements of .

All of the conditions here must be true.

Every element of  must map to one or more elements of .

No element of  may map to multiple elements of .

Every element of  must map to one or more elements of .

Every element of  must map to one or more elements of .

Explanation:

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 .

### Example Question #4 : Relations, Functions And Cartesian Product

For a surjective function  from set  to set  defined by , which of the following does NOT need to be true?

No element of  may map to multiple elements of .

Every element of  must map to one or more elements of .

No element of  may map to multiple elements of .

All of the conditions here must be true.

Every element of  must map to one or more elements of .

No element of  may map to multiple elements of .

Explanation:

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 .

### Example Question #5 : Relations, Functions And Cartesian Product

For which of the following pairs is the cardinality of the two sets equal?

and , where there exists an injective, non-surjective function .

Explanation:

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 .

### Example Question #6 : Relations, Functions And Cartesian Product

What type of function is  where  ?

None of these answers is correct.

Bijective

Injective

Surjective

Surjective

Explanation:

Because multiple elements of  can map to a single element of  (e.g. -2 and 2 map to 2), this function is surjective.

### Example Question #7 : Relations, Functions And Cartesian Product

What type of function is  where  ?

Bijective

None of these answers is correct.

Surjective

Injective

Injective

Explanation:

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.

### Example Question #2 : Relations, Functions And Cartesian Product

What type of function is  where  ?

None of these answers is correct.

Surjective

Injective

Bijective