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Question Number 35703 by scientist last updated on 22/May/18

Answered by arcana last updated on 17/Oct/18

$$\mathrm{tomando}\mathrm{la}\mathrm{propiedad}\mathrm{de}\mathrm{funcion}\mathrm{de}\mathrm{medida}n$$$$\mathrm{A},\mathrm{B}\mathrm{conjuntos}\mathrm{finitos}$$$$n(\mathrm{A}\cup \mathrm{B})=n\left(\mathrm{A}\right)+n\left(\mathrm{B}\right)-n(\mathrm{A}\cap \mathrm{B})$$$$\mathrm{asi}$$$$\bullet n((\mathrm{A}\cup \mathrm{B})\cup (\mathrm{C}\cup \mathrm{D}))=n(\mathrm{A}\cup \mathrm{B})+n(\mathrm{C}\cup \mathrm{D})-n((\mathrm{A}\cup \mathrm{B})\cap (\mathrm{C}\cup \mathrm{D}))$$$$=[n\left(\mathrm{A}\right)+n\left(\mathrm{B}\right)-n(\mathrm{A}\cap \mathrm{B})]+[n\left(\mathrm{C}\right)+n\left(\mathrm{D}\right)-n(\mathrm{C}\cap \mathrm{D})]-n(((\mathrm{A}\cup \mathrm{B})\cap \mathrm{C})\cup ((\mathrm{A}\cup \mathrm{B})\cap \mathrm{D}))$$$$\frac{}{}\frac{}{}\frac{}{}\frac{}{}\frac{}{}\frac{}{}\frac{}{}\frac{}{}\frac{}{}\frac{}{}\frac{}{}\frac{}{}\frac{}{}\frac{}{}\frac{}{}\frac{}{}\frac{}{}\frac{}{}\frac{}{}\frac{}{}\frac{}{}$$$$\mathrm{ahora}\mathrm{hallamos}n([(\mathrm{A}\cup \mathrm{B})\cap \mathrm{C}]\cup [(\mathrm{A}\cup \mathrm{B})\cap \mathrm{D}])$$$$n([(\mathrm{A}\cup \mathrm{B})\cap \mathrm{C}]\cup [(\mathrm{A}\cup \mathrm{B})\cap \mathrm{D}])=n((\mathrm{A}\cup \mathrm{B})\cap \mathrm{C})+$$$$n((\mathrm{A}\cup \mathrm{B})\cap \mathrm{D})-n((\mathrm{A}\cup \mathrm{B})\cap (\mathrm{C}\cap \mathrm{D}))$$$$$$$$=[n(A\cap \mathrm{C})+n(\mathrm{B}\cap \mathrm{C})-n(\mathrm{A}\cap \mathrm{B}\cap \mathrm{C})]+$$$$[n(A\cap \mathrm{D})+n(\mathrm{B}\cap \mathrm{D})-n(\mathrm{A}\cap \mathrm{B}\cap \mathrm{D})]-$$$$[n(A\cap \mathrm{C}\cap \mathrm{D})+n(\mathrm{B}\cap \mathrm{C}\cap \mathrm{D})-n(\mathrm{A}\cap \mathrm{B}\cap \mathrm{C}\cap \mathrm{D})]$$$$\frac{}{}\frac{}{}\frac{}{}\frac{}{}\frac{}{}\frac{}{}\frac{}{}\frac{}{}\frac{}{}\frac{}{}\frac{}{}\frac{}{}\frac{}{}\frac{}{}\frac{}{}\frac{}{}\frac{}{}\frac{}{}\frac{}{}\frac{}{}\frac{}{}$$$$\mathrm{por}\mathrm{lo}\mathrm{anterior}\mathrm{podemos}\mathrm{reescribir}\bullet \mathrm{como}$$$$\bullet n((\mathrm{A}\cup \mathrm{B})\cup (\mathrm{C}\cup \mathrm{D}))=n\left(\mathrm{A}\right)+n\left(\mathrm{B}\right)+n\left(\mathrm{C}\right)+n\left(\mathrm{D}\right)-n(A\cap \mathrm{B})-n(\mathrm{C}\cap \mathrm{D})-$$$$n(A\cap \mathrm{C})-n(\mathrm{B}\cap \mathrm{C})+n(\mathrm{A}\cap \mathrm{B}\cap \mathrm{C})]-$$$$n(A\cap \mathrm{D})-n(\mathrm{B}\cap \mathrm{D})+n(\mathrm{A}\cap \mathrm{B}\cap \mathrm{D})]+$$$$n(A\cap \mathrm{C}\cap \mathrm{D})+n(\mathrm{B}\cap \mathrm{C}\cap \mathrm{D})-n(\mathrm{A}\cap \mathrm{B}\cap \mathrm{C}\cap \mathrm{D})$$$$$$$$$$

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