Number of Onto, Into & Constant Functions from A to B

In summary, the conversation discusses the total number of onto, into, and constant functions from set $A$ to set $B$, where $A = \left\{1,2,3,4,5,6\right\}$ and $B = \left\{a,b,c,d,e\right\}$. For the first question, inclusion/exclusion is used to determine the answer. The second question involves finding the number of functions from a set of six elements to a set of five elements, which is $5^6$. The third question asks for the concept of a constant function.
  • #1
juantheron
247
1
If $A = \left\{1,2,3,4,5,6\right\}$ and $B = \left\{a,b,c,d,e\right\}$. Then Total no. of $(1)$ onto function from $A$ to $B$$(2)$ into function from $A$ to $B$$(3)$ Constant function from $A$ to $B$Plz explain it breifly"Thanks"
 
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  • #2
jacks said:
If $A = \left\{1,2,3,4,5,6\right\}$ and $B = \left\{a,b,c,d,e\right\}$. Then Total no. of
$(1)$ onto function from $A$ to $B$
$(2)$ into function from $A$ to $B$
$(3)$ Constant function from $A$ to $B$

For #1 use inclusion/exclusion to figure the answer.

In #2 how is into used? There are $5^6$ functions from a set six to a set of five.

For #3 think what it means to be a constant function.
 

FAQ: Number of Onto, Into & Constant Functions from A to B

What is the difference between onto, into, and constant functions?

Onto, into, and constant functions are all types of functions that describe the relationship between two sets, A and B. Onto functions are also known as surjective functions and they map all elements in set A to set B, meaning that every element in set B has at least one corresponding element in set A. Into functions, also called injective functions, map each element in set A to a unique element in set B, meaning that no two elements in set A can map to the same element in set B. Constant functions are functions where every element in set A maps to the same element in set B, meaning that the output is always the same for any input.

How do you determine the number of onto functions from set A to set B?

The number of onto functions from set A to set B can be determined using the formula n^m - n^(m-1), where n is the number of elements in set B and m is the number of elements in set A. This formula takes into account the fact that for an onto function, at least one element in set B must have more than one corresponding element in set A.

Is it possible to have an onto function from a smaller set to a larger set?

Yes, it is possible to have an onto function from a smaller set to a larger set. This is because the number of onto functions from set A to set B is dependent on the number of elements in set A and set B, not the size of the sets.

Can a function be both onto and into at the same time?

Yes, a function can be both onto and into at the same time. This type of function is called a bijection and it means that every element in set A has a unique corresponding element in set B, and every element in set B has at least one corresponding element in set A.

How do constant functions differ from other types of functions?

Constant functions differ from other types of functions in that their output is always the same for any input. This means that the function is not dependent on the input and will always produce the same result. In contrast, onto and into functions have specific rules for mapping inputs to outputs, and the output may vary depending on the input.

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