Find the intersections and unions of the given sets.

Eighteen students from a school appear in one or more subjects for an inter-school quiz competition as shown in the table below.

\begin{tabular}{|c|c|c|}
\hline \begin{tabular}{c}
General \\
Knowledge
\end{tabular} & Math & Science \\
\hline Acel & Barek & Carlin \\
\hline Acton & Bay & Acton \\
\hline Anael & Max & Anael \\
\hline Max & Kai & Kai \\
\hline Carl & Anael & Dario \\
\hline Dario & Carlin & Barek \\
\hline
\end{tabular}

Let [tex]$G$[/tex] represent the set of students appearing for General Knowledge, [tex]$M$[/tex] represent the set of students appearing for Math, and [tex]$S$[/tex] represent the set of students appearing for Science.

Find [tex]$G \cap M$[/tex] and [tex]$G \cup S$[/tex].



Answer :

Certainly! Let's solve this step by step. We are given the sets of students appearing for each subject:

General Knowledge (G):
- Acel
- Acton
- Anael
- Max
- Carl
- Dario

Math (M):
- Barek
- Bay
- Max
- Kai
- Anael
- Carlin

Science (S):
- Carlin
- Acton
- Anael
- Kai
- Dario
- Barek

We need to find the intersections and unions of the sets as described.

### Finding [tex]\( G \cap M \)[/tex]
The intersection of two sets, [tex]\( G \cap M \)[/tex], includes elements that are present in both sets G and M.

- Elements in [tex]\( G \)[/tex]: {Acel, Acton, Anael, Max, Carl, Dario}
- Elements in [tex]\( M \)[/tex]: {Barek, Bay, Max, Kai, Anael, Carlin}

Now, identify the common elements between G and M:
- Anael is present in both G and M.
- Max is present in both G and M.

So,
[tex]\[ G \cap M = \{ \text{Anael, Max} \} \][/tex]

### Finding [tex]\( G \cup S \)[/tex]
The union of two sets, [tex]\( G \cup S \)[/tex], includes all unique elements present in either set G or S.

- Elements in [tex]\( G \)[/tex]: {Acel, Acton, Anael, Max, Carl, Dario}
- Elements in [tex]\( S \)[/tex]: {Carlin, Acton, Anael, Kai, Dario, Barek}

Now, combine all unique elements from G and S:
- We list all the elements from [tex]\( G \)[/tex]: Acel, Acton, Anael, Max, Carl, Dario
- We then include the additional unique elements from [tex]\( S \)[/tex]: Carlin, Kai, Barek

So,
[tex]\[ G \cup S = \{ \text{Acel, Acton, Anael, Max, Carl, Dario, Carlin, Kai, Barek} \} \][/tex]

Combining everything, the final answers are:

- [tex]\( G \cap M = \{ \text{Max, Anael} \} \)[/tex]
- [tex]\( G \cup S = \{ \text{Acel, Acton, Anael, Max, Carl, Dario, Carlin, Kai, Barek} \} \)[/tex]