In three sets Q, R and S, the following conditions are found.
In Q R ∩ there are 9 elements.
In Q S ∩ there are 4 elements.
In R S ∩ there are 11 elements.
If (QR QS RS ∩ ∪ ∩∪ ∩ ) ( ) ( ) includes 18 elements, how many
elements are in QRS ∩ ∩ ?
I've never seen reverse Polish notation used like this.
Most people write R∩S instead of RS∩
Your notation makes it quite unclear what
QR QS RS ∩ ∪ ∩∪ ∩ ) ( ) ( )
means.
In three sets Q, R, and S, the following conditions are found: in Q intersect R. there are 9 elements in Q intersect S, there are 4 elements in R intersect S, there are 11 elements if (Q Intersect R) union (Q Intersect S) union (R intersect S) includes 18 elements, how many elements are in Q intersect R intersect S?
Well, it seems like someone has thrown me a mathematical riddle. Let me put on my thinking cap, which is actually a sparkly top hat. 🎩
Let's break this down step by step, shall we?
We know that Q R ∩ has 9 elements, Q S ∩ has 4 elements, and R S ∩ has 11 elements. So, let's add these up and see what we get: 9 + 4 + 11 = 24.
Now, the expression (QR QS RS ∩ ∪ ∩∪ ∩ ) ( ) ( ) represents the intersection and union of these sets. Since we're dealing with three sets (Q, R, and S), the expression can be simplified as follows: (Q ∩ R ∩ S) ∪ (Q ∩ R) ∪ (Q ∩ S) ∪ (R ∩ S).
We're told that this expression contains 18 elements. So, let's set up an equation: 18 = (Q ∩ R ∩ S) + (Q ∩ R) + (Q ∩ S) + (R ∩ S).
Hmmm, we have a problem here. We can't deduce the exact values of Q ∩ R ∩ S, Q ∩ R, Q ∩ S, and R ∩ S from the given information. Therefore, we can't determine how many elements are in the intersection of QRS.
But don't worry, my friend! Even though I couldn't provide you with a straightforward answer, I hope I at least brought a smile to your face. Remember, sometimes the journey is more enjoyable than the destination! 🤡
To find the number of elements in QRS ∩ ∩ (the intersection of all three sets Q, R, and S), we need to use the principle of inclusion-exclusion.
Let's break down the problem step-by-step:
1. Start with the sum of the sizes of each individual set: |Q| + |R| + |S|.
- This represents the total number of elements in sets Q, R, and S without considering any intersections.
2. Subtract the size of Q ∩ R since we have double-counted it in step 1.
- Equation becomes |Q| + |R| + |S| - |Q ∩ R|.
3. Subtract the size of Q ∩ S since we have double-counted it in step 1.
- Equation becomes |Q| + |R| + |S| - |Q ∩ R| - |Q ∩ S|.
4. Subtract the size of R ∩ S since we have double-counted it in step 1.
- Equation becomes |Q| + |R| + |S| - |Q ∩ R| - |Q ∩ S| - |R ∩ S|.
5. Add the size of (Q ∩ R ∩ S) since it was subtracted twice (once in step 2 and again in step 3).
- Equation becomes |Q| + |R| + |S| - |Q ∩ R| - |Q ∩ S| - |R ∩ S| + |Q ∩ R ∩ S|.
Given the information provided:
|Q ∩ R| = 9
|Q ∩ S| = 4
|R ∩ S| = 11
|(Q ∩ R ∩ S)| = 18 (as mentioned in the question)
Substituting these values into the equation:
|Q| + |R| + |S| - 9 - 4 - 11 + 18
Simplifying the equation, we get:
|Q| + |R| + |S| - 6
Therefore, the number of elements in QRS ∩ ∩ is equal to |Q| + |R| + |S| - 6.
To find the number of elements in the intersection of sets Q, R, and S (QRS ∩ ∩), we can use the Principle of Inclusion-Exclusion.
We start by finding the number of elements in the union of the three sets, which is given to be 18 elements: (QR QS RS ∩ ∪ ∩∪ ∩ ) ( ) ( ) = 18.
Step 1: Find the total number of elements in each individual set (Q, R, and S).
- Let n(Q) represent the number of elements in set Q.
- Let n(R) represent the number of elements in set R.
- Let n(S) represent the number of elements in set S.
Step 2: Find the total number of elements in the intersection of each pair of sets.
- Let n(Q ∩ R) represent the number of elements in the intersection of sets Q and R.
- Let n(Q ∩ S) represent the number of elements in the intersection of sets Q and S.
- Let n(R ∩ S) represent the number of elements in the intersection of sets R and S.
Step 3: Use the Principle of Inclusion-Exclusion formula to calculate the number of elements in the union of the three sets.
n(Q ∪ R ∪ S) = n(Q) + n(R) + n(S) - n(Q ∩ R) - n(Q ∩ S) - n(R ∩ S) + n(QRS ∩)
From the given information:
n(Q ∩ R) = 9
n(Q ∩ S) = 4
n(R ∩ S) = 11
Substituting these values into the formula:
18 = n(Q) + n(R) + n(S) - 9 - 4 - 11 + n(QRS ∩)
Simplifying the equation:
18 = n(Q) + n(R) + n(S) - 24 + n(QRS ∩)
n(Q) + n(R) + n(S) + n(QRS ∩) = 18 + 24
n(Q) + n(R) + n(S) + n(QRS ∩) = 42
So, the number of elements in the intersection of sets Q, R, and S (QRS ∩ ∩) is 42.