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]
