Example Question - venn diagram
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Determining the Number of Teachers Teaching Physics
<p>Let $P$ be the number of teachers who teach physics and $M$ be the number of teachers who teach mathematics.</p> <p>We are given that there are 20 teachers in total, of which 12 teach mathematics, 4 teach both physics and mathematics. We need to find how many teach physics.</p> <p>We can use the principle of inclusion and exclusion to find the number of teachers who teach only physics.</p> <p>Number of teachers teaching only mathematics is $M - 4$.</p> <p>Number of teachers teaching physics, including those who teach both subjects, is $P$.</p> <p>The sum of teachers teaching only mathematics, only physics, and both is the total number of teachers:</p> <p>$(M - 4) + P = 20$</p> <p>Given that $M = 12$, we substitute this into the equation:</p> <p>$(12 - 4) + P = 20$</p> <p>$8 + P = 20$</p> <p>$P = 20 - 8$</p> <p>$P = 12$</p> <p>There are 12 teachers who teach physics.</p>
Analyzing Student Sport Preferences
<p>(a) Untuk mewakilkan setiap murid dengan huruf pertama dari nama mereka:</p> <p>M = \{ Badminton, Tennis \}</p> <p>S = \{ Badminton, Tennis, Hockey \}</p> <p>E = \{ Badminton, Tennis \}</p> <p>F = \{ Badminton, Tennis, Hockey \}</p> <p>B = \{ Badminton, Tennis \}</p> <p>C = \{ Hockey \}</p> <p>G = \{ Hockey \}</p> <p>H = \{ Hockey, Tennis \}</p> <p>Lukis diagram Venn untuk menunjukkan hubungan antara set-set ini.</p> <p>(b) Menggunakan simbol "{u}" untuk menggambarkan daerah yang diduduki oleh Davis dan Franco:</p> <p>D \cup F = \{ Badminton, Tennis, Hockey \}</p> <p>Lukis diagram Venn dan tunjukkan daerah D \cup F dengan simbol "{u}".</p>
Analysis of Students' Sports Preferences
<p>\textbf{(a) Senaraikan nama murid yang meminati setidaknya satu jenis sukan dengan menggunakan takat-takat set.} </p> <p> \text{Guna huruf pertama daripada nama bagi mewakili setiap murid.} \</p> <p> \text{Let } A = \text{Badminton, } B = \text{Tennis, } C = \text{Hockey.} \</p> <p> A \cup B \cup C = \{M, E, A, F, C, G, H, S\} </p> <p>\textbf{(b) Nyatakan setiap elemen dalam } P(A \cup B \cup C) \textbf{ yang memuatkan bilangan anggota tiga (3) sahaja.} </p> <p> P(A \cup B \cup C) \text{ with exactly 3 members:} </p> <p> \{M, E, A\}, \{M, E, F\}, \{M, E, C\}, \{M, E, G\}, \{M, E, H\}, \{M, E, S\}, \{M, A, F\}, \{M, A, C\}, \{M, A, G\}, \{M, A, H\}, ... (and so on, list all possible combinations with exactly 3 elements from the union set) </p> <p> \text{Since there are many possible combinations, only a partial list is provided here. Complete the list to contain all unique combinations of exactly 3 members from the set } A \cup B \cup C. </p> <p>\textbf{(c) Tanpa menggunakan set Venn, nyatakan himpunan bahagian } P(A \cup B \cup C) \textbf{ yang memuatkan sekurang-kurangnya satu murid yang meminati badminton.} </p> <p> \text{Let } A = \text{Badminton.} </p> <p> \text{The power set of } A \cup B \cup C \text{ containing at least one member who is interested in badminton (the set A) includes all subsets that contain the elements of A. Since A includes some of the same members as B and C, the power set would include subsets such as: } \{M\}, \{M, E\}, \{M, A\}, \{M, F\}, \{M, C\}, ... (and so on, complete with all subsets including at least one element of A). </p> <p> \text{Again, the complete set of subsets is extensive, thus only a partial list is shown. The complete solution would list all subsets of the power set containing at least one member from set A.} </p>
Finding the GCD and LCM of Numbers
Zur Lösung dieser Aufgabe müssen wir das kleinste gemeinsame Vielfache (kgV) und den größten gemeinsamen Teiler (ggT) der Zahlen 18, 60 und 50 bestimmen. Beginnen wir mit dem ggT: 1. Zerlegen wir jede Zahl in ihre Primfaktoren. - \( 18 = 2 \times 3^2 \) - \( 60 = 2^2 \times 3 \times 5 \) - \( 50 = 2 \times 5^2 \) 2. Der ggT ist das Produkt der gemeinsamen Primfaktoren, genommen mit dem niedrigsten Exponenten, der in allen Zahlen vorkommt. - Der gemeinsame Primfaktor von 18, 60 und 50 ist 2, der mit dem niedrigsten Exponenten einmal vorkommt. - ggT(18, 60, 50) = 2 Nun zum kgV: 1. Das kgV ist das Produkt aller Primfaktoren, die in irgendeiner der Zahlen vorkommen, wobei jeder Faktor mit dem höchsten Exponenten, der in irgendeiner der Zahlen vorkommt, genommen wird. - Wir haben als Primfaktoren 2, 3 und 5. - Der höchste Exponent für 2 ist 2 (in 60), für 3 ist 2 (in 18) und für 5 ist 2 (in 50). - kgV(18, 60, 50) = \( 2^2 \times 3^2 \times 5^2 \) = \( 4 \times 9 \times 25 \) = 900 Um diese Werte in einem Venn-Diagramm zu markieren, würde man eine Kreisgruppe für jede der drei Zahlen zeichnen, wobei der Schnittpunkt aller drei Kreise (der gemeinsame Bereich) die Zahl 2 (den ggT) enthält, und man würde außen an einer Seite, die alle drei Kreise verbindet, das kgV (900) platzieren.
Finding Largest Possible Number of Items in Set F
The image shows a Venn diagram with two sets, F and G, intersecting. Within set F, but not in the intersection, the expression "9u + 12" is written. Within set G, but not in the intersection, the expression "8u + 3" is written. Within the intersection, the expression "7u + 3" is given. The question asks: Given that there are fewer than 94 items in total, what is the largest possible number of items in set F? In order to find the largest possible number of items in set F, considering there are fewer than 94 items in total, we will first express the total number of items combining both sets, F and G, including their intersection. The total number of items in set F is the sum of the distinct items in F and the items F shares with G, which would be (9u + 12) + (7u + 3). The total number of items in set G is the sum of the distinct items in G and the items G shares with F, which would be (8u + 3) + (7u + 3). Since the sets F and G overlap in the intersection, we should not double count the intersection when calculating the grand total. Hence, the intersection (7u + 3) is added just once to the grand total. So the total number of items is: (9u + 12) + (8u + 3) + (7u + 3) - (7u + 3) The `7u + 3` from the intersection is added twice but subtracted once to account correctly for those items that are both in F and G. Simplifying, the total is: (9u + 12) + (8u + 3) + (7u + 3) - (7u + 3) = 9u + 12 + 8u + 3 = 17u + 15 Now, we use the information given in the question that there are fewer than 94 items in total: 17u + 15 < 94 Subtracting 15 from both sides, we get: 17u < 79 Dividing by 17, we find: u < 79 / 17 u < 4.647... Since u must be a whole number (as it represents the count of items), the largest possible value for u is 4. Now, we can find the largest possible number of items in set F by plugging in u = 4 into our expression for set F: 9u + 12 = 9(4) + 12 = 36 + 12 = 48 Therefore, the largest possible number of items in set F, given that there are fewer than 94 items in total, is 48.
Completing a Venn Diagram with Given Set Information
This Venn diagram represents three sets: A, B, and C, and their intersections within the universal set S. We are given the following information: - The number of elements in set A is n(A) = 8. - The number of elements in set B is n(B) = 16. - The number of elements in set S is n(S) = 135. Let's use this information to fill in the missing parts of the Venn diagram. 1. First, sum up the numbers already given inside the Venn diagram to find the number of elements in set C: Inside C (but outside A and B) = 15 Inside C and A (but outside B) = 3 Inside C and B (but outside A) = 10 Inside C, A, and B = 4 Total so far = 15 + 3 + 10 + 4 = 32 2. Since we don't have n(C) given explicitly, we cannot directly fill the value for C yet. Instead, let's consider set A and set B and their overlap. For A, apart from the 3 (exclusive to A and C) and the 4 (in the intersection of A, B, and C), there must be another 1 element which is exclusive to A (since 8 elements are in A in total): 3 + 4 + 1 = 8. Now, let's update the diagram's inside A but outside B and C to be 1. 3. Moving on to B, there is already 4 inside the intersection with A and C, and 10 in the intersection with C, meaning we need to find how many elements are exclusively in B by subtracting the known elements inside B from the total number in B, n(B)= 16. Total number in B = 16, and we know the intersections: Inside B and A = 4 Inside B and C = 10 Exclusive to B = 16 - (Intersection B and A + Intersection B and C) = 16 - (4 + 10) = 2. Now, let's update the diagram's inside B but outside A and C to be 2. 4. Now, we can find out the number of elements exclusive to C by subtracting the known elements in intersections involving C from what we've calculated for set C: Exclusive to C = (Total so far in C) - (Inside C and A) - (Inside C and B) - (Inside C, A, and B) Exclusive to C = 32 - 3 - 10 - 4 = 15. But this value was already filled in the diagram within C but outside A and B, so our previous calculations are consistent. 5. Lastly, fill in the value of the universal set S, which includes all the elements in sets A, B, C, and also those that are not in any of these sets: Total elements in the sets A, B, and C (already filled in diagram including all intersections) = 1 + 3 + 4 + 10 + 2 + 15 = 35. Hence, the number of elements not in sets A, B, or C but within the universal set S = n(S) - total in A, B, C = 135 - 35 = 100. This last value of 100 is also already present in the diagram, which indicates the number of elements outside sets A, B, and C but within S. With all numbers filled in the Venn diagram, each subset within the universal set is accounted for.
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