Himpunan

Jika S adalah himpunan semesta, A dan B adalah himpunan bagian dari S, maka: $$n(S)=n(A)+n(B)-n(A\cap B)+n(A\cup B{)}^{\prime }$$ Bukti:

b=n(A\cap B)

$$\begin{array}{rl}a+b& =n(A)\\ a+n(A\cap B)& =n(A)\\ a& =n(A)-n(A\cap B)\end{array}$$

$$\begin{array}{rl}b+c& =n(B)\\ n(A\cap B)+c& =n(B)\\ c& =n(B)-n(A\cap B)\end{array}$$

d=n(A\cup B)'

$$\begin{array}{rl}n(S)& =a+b+c+d\\ & =n(A)-n(A\cap B)+n(A\cap B)+n(B)-n(A\cap B)+n(A\cup B{)}^{\prime }\\ & =n(A)+n(B)-n(A\cap B)+n(A\cup B{)}^{\prime }\end{array}$$

Jika S adalah himpunan semesta, A, B dan C adalah himpunan bagian dari S, maka: $$n(S)=n(A)+n(B)+n(C)-n(A\cap B)-n(B\cap C)-n(A\cap C)+n(A\cap B\cap C)+n(A\cup B\cup C{)}^{\prime }$$ Bukti:

g=n(A\cap B\cap C)

$$\begin{array}{rl}d+g& =n(A\cap B)\\ d+n(A\cap B\cap C)& =n(A\cap B)\\ d& =n(A\cap B)-n(A\cap B\cap C)\end{array}$$

$$\begin{array}{rl}e+g& =n(A\cap C)\\ e+n(A\cap B\cap C)& =n(A\cap C)\\ e& =n(A\cap C)-n(A\cap B\cap C)\end{array}$$

$$\begin{array}{rl}f+g& =n(B\cap C)\\ f+n(A\cap B\cap C)& =n(B\cap C)\\ f& =n(B\cap C)-n(A\cap B\cap C)\end{array}$$

$$\begin{array}{rl}a+d+e+g& =n(A)\\ a+n(A\cap B)-n(A\cap B\cap C)+n(A\cap C)-n(A\cap B\cap C)+n(A\cap B\cap C)& =n(A)\\ a+n(A\cap B)+n(A\cap C)-n(A\cap B\cap C)& =n(A)\\ a& =n(A)-n(A\cap B)-n(A\cap C)+n(A\cap B\cap C)\end{array}$$

$$\begin{array}{rl}b+d+f+g& =n(B)\\ b+n(A\cap B)-n(A\cap B\cap C)+n(B\cap C)-n(A\cap B\cap C)+n(A\cap B\cap C)& =n(B)\\ b+n(A\cap B)+n(B\cap C)-n(A\cap B\cap C)& =n(B)\\ b& =n(B)-n(A\cap B)-n(B\cap C)+n(A\cap B\cap C)\end{array}$$

$$\begin{array}{rl}c+e+f+g& =n(C)\\ c+n(A\cap C)-n(A\cap B\cap C)+n(B\cap C)-n(A\cap B\cap C)+n(A\cap B\cap C)& =n(C)\\ c+n(A\cap C)+n(B\cap C)-n(A\cap B\cap C)& =n(C)\\ c& =n(C)-n(A\cap C)-n(B\cap C)+n(A\cap B\cap C)\end{array}$$

h=n(A\cup B\cup C)'

$$\begin{array}{rl}n(S)& =a+b+c+d+e+f+g+h\\ & =n(A)-n(A\cap B)-n(A\cap C)+n(A\cap B\cap C)+n(B)-n(A\cap B)-n(B\cap C)+n(A\cap B\cap C)+n(C)-n(A\cap C)-n(B\cap C)+n(A\cap B\cap C)+n(A\cap B)-n(A\cap B\cap C)+n(A\cap C)-n(A\cap B\cap C)+n(B\cap C)-n(A\cap B\cap C)+n(A\cap B\cap C)+n(A\cup B\cup C{)}^{\prime }\\ & =n(A)+n(B)+n(C)-n(A\cap B)-n(B\cap C)-n(A\cap C)+n(A\cap B\cap C)+n(A\cup B\cup C{)}^{\prime }\end{array}$$