In a certain class, 22 pupils take one or more of chemistry, Economics and Government.12 take Economics (E),8 take Government (G) and 7take Chemistry (C). Nobody takes Economics and chemistry and 4 pupils take Economics and Government.
I) a) using set notation and the letters indicated above, write down the two statements in the last sentence
b) draw a Venn diagram to illustrate the information.
II) how many pupils take
a) both chemistry and government?
b) government only?
I) a) E ∩ C = ∅
b) E ∩ G = 4
II)
a) Let x be the number of pupils taking both chemistry and government. Then from the Venn diagram, we can set up the equation:
x + 4 + (8 - x) + (7 - x) + (12 - 4 - x) = 22
Simplifying and solving for x, we get:
x = 1
Therefore, only 1 pupil takes both chemistry and government.
b) From the Venn diagram, we can see that the number of pupils taking government only is the number in the G circle that is not shared with any other circle. This is (8 - x) = 7, since we found that x = 1 in part a). Therefore, 7 pupils take government only.
I)
a) The two statements in the last sentence can be written in set notation as:
1) E ∩ C = ∅ (No one takes Economics and Chemistry)
2) E ∩ G = 4 (4 pupils take Economics and Government)
b) The Venn diagram representing the given information would have three overlapping circles labeled E, G, and C. The circles would intersect each other as follows:
_____
| |
| E |
|_____|
/ \
/ \
___/_____________\___
| | |
| G | C |
|___________|___________|
II)
a) To determine the number of students who take both Chemistry and Government, we need to find the intersection of the sets C and G. From the given information, it is not explicitly stated how many students take both Chemistry and Government. Therefore, we cannot determine the exact number.
b) To find the number of students who take Government only, we need to subtract the number of students who take both (from part a) and the number of students who take both Economics and Government (as given). Therefore, the number of students who take Government only is:
8 - 4 = 4 students.