MHB How many subsets of set A satisfy given conditions?

  • Thread starter Thread starter Evgeny.Makarov
  • Start date Start date
  • Tags Tags
    Finite Sets
Evgeny.Makarov
Gold Member
MHB
Messages
2,434
Reaction score
4
Consider a set $A$ and its subsets $B$ and $C$. It is known that $|A-(B\cap C)|=8$, $|B|=5$, $|C-B|=1$ and $|B\cap C|=3$ (here $-$ denotes set difference). How many subsets $X\subseteq A$ are there if $X\cap B\cap C\ne\emptyset$, $|X-(B\cup C)|\ge3$ and $|X\cap (B-C)|=2$?
 
Mathematics news on Phys.org
Since nobody gave an answer, I am posting a solution.

I will omit $\cap$ and write intersection as multiplication. I will also write $\sqcup$ for disjoint union.

In this problem $A$ is the universal set, and each element of $A$ belongs to exactly one of four classes: it can fall in or outside $B$ and $C$. Since $A-BC=\overline{BC}=\overline{B}\cup\overline{C}=\overline{B}C\sqcup B\overline{C}\sqcup\overline{B}\overline{C}$, $B=BC\sqcup B\overline{C}$ and $C-B=\overline{B}C$, we have the following system of equations.
\begin{align*}
|\overline{B}C|+|B\overline{C}|+|\overline{B}\overline{C}|&=8\\
|BC|+|B\overline{C}|&=5\\
|\bar{B}C|&=1\\
|BC|&=3
\end{align*}
Solving this system, we can fill the following table form of the Venn diagram.
\begin{tikzpicture}[scale=1.3,y={(0cm,-1cm)}]
\draw[step=1cm] (-.3,-.3) grid (2,2);
\node[above] at (.5,0) {$B$};
\node[above] at (1.5,0) {$\overline{B}$};
\node
at (0,.5) {$C$};
\node
at (0,1.5) {$\overline{C}$};
\path[shift={(.5,.5)}] (0,0) node {3} (1,0) node {1} (0,1) node {2} (1,1) node {5};
\end{tikzpicture}

Set \(X\) is also split into four disjoint classes. The condition $XBC\ne\emptyset$ gives us $2^3-1=7$ variants for $XBC$. The condition $|X-(B\cup C)|\ge3$ means that $X\overline{B}\overline{C}$ can be chosen in $\binom{5}{3}+\binom{5}{4}+\binom{5}{5}=16$ ways. The condition $|X(B-C)|=|XB\overline{C}|=2$ requires that both elements of $B\overline{C}$ are included in $X$. Finally, the problem does not say anything about $X\overline{B}C$, which gives us two variants: the only element of $\overline{B}C$ may or may not be in $X$. Altogether, there are $7\cdot16\cdot2=224$ variants for $X$.​


I would appreciate your opinion on the difficulty level of this problem. Which year in the university is it appropriate for?​
 
That looks like a fairly standard algebra or pre-calculus problem. I wouldn't be surprised to see it in a college freshman or even secondary school class.
 
Insights auto threads is broken atm, so I'm manually creating these for new Insight articles. In Dirac’s Principles of Quantum Mechanics published in 1930 he introduced a “convenient notation” he referred to as a “delta function” which he treated as a continuum analog to the discrete Kronecker delta. The Kronecker delta is simply the indexed components of the identity operator in matrix algebra Source: https://www.physicsforums.com/insights/what-exactly-is-diracs-delta-function/ by...
Fermat's Last Theorem has long been one of the most famous mathematical problems, and is now one of the most famous theorems. It simply states that the equation $$ a^n+b^n=c^n $$ has no solutions with positive integers if ##n>2.## It was named after Pierre de Fermat (1607-1665). The problem itself stems from the book Arithmetica by Diophantus of Alexandria. It gained popularity because Fermat noted in his copy "Cubum autem in duos cubos, aut quadratoquadratum in duos quadratoquadratos, et...
I'm interested to know whether the equation $$1 = 2 - \frac{1}{2 - \frac{1}{2 - \cdots}}$$ is true or not. It can be shown easily that if the continued fraction converges, it cannot converge to anything else than 1. It seems that if the continued fraction converges, the convergence is very slow. The apparent slowness of the convergence makes it difficult to estimate the presence of true convergence numerically. At the moment I don't know whether this converges or not.
Back
Top