If f: A → B is a function and |A| = 5, |B| = 3, what is the maximum number of di
Practice Questions
Q1
If f: A → B is a function and |A| = 5, |B| = 3, what is the maximum number of distinct functions that can be formed?
3^5
5^3
15
8
Questions & Step-by-Step Solutions
If f: A → B is a function and |A| = 5, |B| = 3, what is the maximum number of distinct functions that can be formed?
Step 1: Understand what a function is. A function f: A → B means that every element in set A is assigned to exactly one element in set B.
Step 2: Identify the sizes of the sets. Here, |A| = 5 means there are 5 elements in set A, and |B| = 3 means there are 3 elements in set B.
Step 3: Realize that for each element in set A, you can choose any of the elements in set B. Since there are 3 choices in B for each of the 5 elements in A, you will multiply the choices.
Step 4: Calculate the total number of distinct functions. Since each of the 5 elements in A can map to any of the 3 elements in B, the total number of functions is 3 (choices for the first element) multiplied by 3 (choices for the second element) multiplied by 3 (choices for the third element) multiplied by 3 (choices for the fourth element) multiplied by 3 (choices for the fifth element). This is the same as 3 raised to the power of 5.
Step 6: Conclude that the maximum number of distinct functions that can be formed is 243.
Function Mapping – Understanding how a function maps elements from one set (domain) to another set (codomain).
Cardinality – The concept of cardinality refers to the number of elements in a set, which is crucial for determining the number of possible functions.
Exponential Growth of Functions – The number of distinct functions from set A to set B is calculated using the formula |B|^|A|, illustrating exponential growth based on the sizes of the sets.
