Let A and b be two sets with A n and b m how many functions are there from a to b
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The cardinality of $B^A$ is the same if $A$ (resp. $B$) is replaced with a set containing the same number of elements as $A$ (resp. $B$). Show Set $b = |B$|. When $b \lt 2$ there is little that needs to be addressed, so we assume $b \ge 2$. Assume $|A| = n$. A well known result of elementary number theory states that if $a$ is a natural number and $0 \le a \lt b^n$ then it has one and only one base-$\text{b}$ representation, $$\tag 1 a = \sum_{k=0}^{n-1} x_k\, b^k \text{ with } 0 \le x_k \lt b$$ Associate to every $a$ in the initial integer interval $[0, b^n)$ the set of ordered pairs $$\tag 2 \{(k,x_k) \, | \, 0 \le k \lt n \text{ and the base-}b \text{ representation of } a \text{ is given by (1)}\}$$ This association is a
bijective enumeration of $[0, b^n)$ onto the set of all functions Since $[0, b^n)$ has $b^n$ elements, we know how to count all the functions from one finite set into another. How many relations are there between A and B?Since any subset of A × B is a relation from A to B, it follows that if A and B are finite sets then the number of relations from A to B is 2|A×B| = 2|A|·|B|. One way to see this is as the number of subsets of A × B.
How many elements are there in a ∩ B?The intersection A ∩ B has 5 elements.
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. How many functions are there in math?The types of functions can be broadly classified into four types. Based on Element: One to one Function, many to one function, onto function, one to one and onto function, into function. Based on Domain: Algebraic Functions, Trigonometry functions, logarithmic functions.
How many Injective functions are there from A to B?(n−r)!. ∴ There are 24 ways of mapping an injective function from A to B.
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