函数、定义域与值域

\begin{align*} A &= {1,2,3,4,5}\ B&={a,b,c,d,e,g} \end{align*}

f_1 = {(1,c),(2,c),(3,g),(4,g),(5,g)}

\begin{align*} f_2 &= {(1,b),(2,c),(3,d),(4,e),(5,g)} \ f_3 &={(1,a),(2,b),(3,c),(4,d),(5,e)} \ f_4 &= {(1,a),(2,b),(3,b),(4,d),(5,e)} \end{align*}

g_1 = {(1,a),(2,c),(3,d),(5,e)}

\begin{align*} g_2 &= {(1,a),(2,b),(3,b),(3,c),(4,d),(5,e)} \ g_3 &= {(1,a),(1,c),(3,d),(3,e),(3,g)} \ g_4 &= {(1,a),(2,b),(3,b),(4,d),(5,e),(5,g)} \end{align*}

f_5 = {(a, 1),(b, 1),(c, 2),(d, 2),(e, 3),(g, 3)}

\begin{align*} f_6 &= {(a,1),(b,2),(c,3),(d,4),(e,5),(g,3)} \ f_7 &= {(a, 2),(b, 3),(c, 4),(d, 5),(e, 2),(g, 1)}\ f_8 &= {(a, 4),(b, 1),(c, 5),(d, 2),(e, 4),(g, 3)} \end{align*}

\begin{align*} f_9 &= {(g,h),(h,k),(k,m),(m,p),(p,g)} \ f_{10} &= {(g,p),(h,m),(k,k),(m,h),(p,g)} \ f_{11} &= {(g,h),(h,m),(k,h),(m,h),(p,m)} \ f_{12} &= {(g,p),(h,p),(k,g),(m,g),(p,h)} \end{align*}

1def is_function(domain, codomain, function):
2    # Check that the image of every element in the domain is defined
3    for element in domain:
4        # Check if there exists an image of the element
5        if not any(pair[0] == element for pair in function):
6            # If image of element does not exist, return False 
7            return False
8    
9    # Check if each element of the domain has only one image
10    preimages = set()
11    for pair in function:
12        if pair[0] in preimages:
13            return False
14        preimages.add(pair[0])
15    
16    # Check that the range is a subset of the codomain
17    for pair in function: # Pair = (element, image)
18        # Check if every element has an image in the codomain 
19        if pair[1] not in codomain:
20            # If image of any element is not in the codomain, return False
21            return False  
22    return True
23
24# Create a domain set
25domain = {1, 2, 3}
26# Create a codomain set
27codomain = {4, 5, 6}
28# Creating a relation 
29function = {(1, 4), (2, 4), (3, 6)}  
30is_a_function = is_function(domain, codomain, function)
31# Relation is a function or not
32if is_a_function:
33    print("It is a function.")
34else:
35    print("It is not a function.")