Set Theory Operations: A Step-by-Step Guide

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Set Theory Operations: A Step-by-Step Guide

Hey math whizzes! Today, we're diving deep into the fascinating world of set theory. You know, those collections of distinct objects that mathematicians love to play with. We've got a specific problem here involving three sets: P, R, and M, and we need to calculate ni(MP) - R. Sounds a bit intimidating, right? Don't sweat it, guys! We're going to break this down piece by piece, making sure you understand every single step. By the end of this, you'll be a set theory pro, ready to tackle any similar problem that comes your way. Let's get this party started!

Understanding the Sets: P, R, and M

Before we can start crunching numbers and performing operations, we need to get a crystal-clear understanding of what each of our sets represents. Think of these sets as unique boxes, each containing specific items based on certain rules. We've got Set P, Set R, and Set M. Let's unpack them one by one.

Set P: The Natural Numbers Between 0 and 4

First up is Set P. It's defined as P = (3x + 5/x € N; 0 < x < 4). Let's decode this. The x € N part means that 'x' has to be a natural number. Natural numbers are your basic counting numbers: 1, 2, 3, 4, and so on. The condition 0 < x < 4 tells us that 'x' must be greater than 0 and less than 4. So, the possible natural numbers for 'x' that fit this condition are 1, 2, and 3. Now, the 3x + 5 part is the rule for generating the elements of set P. For each valid 'x', we plug it into 3x + 5 to get an element for our set.

  • When x = 1: 3(1) + 5 = 3 + 5 = 8. So, 8 is an element of P.
  • When x = 2: 3(2) + 5 = 6 + 5 = 11. So, 11 is an element of P.
  • When x = 3: 3(3) + 5 = 9 + 5 = 14. So, 14 is an element of P.

Therefore, Set P = {8, 11, 14}. Pretty straightforward, right? We just followed the rules and generated the members.

Set R: Natural Numbers from 1 to 3

Next, let's look at Set R. It's defined as R = (5x/x € N; 1 <= x <= 3). Again, x € N means 'x' is a natural number. The condition 1 <= x <= 3 means 'x' must be greater than or equal to 1 AND less than or equal to 3. The natural numbers that satisfy this are 1, 2, and 3. The rule for generating elements here is 5x. Let's plug in our values for 'x':

  • When x = 1: 5(1) = 5. So, 5 is an element of R.
  • When x = 2: 5(2) = 10. So, 10 is an element of R.
  • When x = 3: 5(3) = 15. So, 15 is an element of R.

Thus, Set R = {5, 10, 15}. We're halfway there, guys!

Set M: A Predefined List of Numbers

Finally, we have Set M. This one is much simpler! It's given directly as M = {4, 5, 8, 9, 11}. No calculations needed here; these are the elements of M as is. It's important to note that sets don't care about the order of elements, and duplicate elements are usually ignored (though not relevant here). So, M contains the numbers 4, 5, 8, 9, and 11.

Performing Set Operations: M ∩ P

Now that we have our sets clearly defined – P = {8, 11, 14}, R = {5, 10, 15}, and M = {4, 5, 8, 9, 11} – we can start performing the required operations. The problem asks for ni(MP) - R. The notation MP is a bit ambiguous without further context, but in standard set theory notation, when two sets are written next to each other like MP, it usually implies the intersection of the two sets, denoted as M ∩ P. The ni part is also unusual; it might refer to the number of elements (cardinality), but let's assume for now it refers to the resulting set from the operation M ∩ P.

So, let's find the intersection of M and P (M ∩ P). The intersection of two sets contains only the elements that are common to both sets. Let's compare our sets:

  • M = {4, 5, 8, 9, 11}
  • P = {8, 11, 14}

Which elements appear in both M and P? Let's check:

  • Is 4 in P? No.
  • Is 5 in P? No.
  • Is 8 in P? Yes!
  • Is 9 in P? No.
  • Is 11 in P? Yes!
  • Is 14 in M? No.

The common elements are 8 and 11. Therefore, M ∩ P = {8, 11}.

The Final Step: (M ∩ P) - R

We're almost at the finish line, folks! The final operation required is (M ∩ P) - R. This symbol '-' represents the set difference. The set difference A - B contains all the elements that are in set A but not in set B.

In our case, we want to find the elements that are in the set (M ∩ P) but not in set R.

  • We found that M ∩ P = {8, 11}.
  • And we know that R = {5, 10, 15}.

Now, let's see which elements from {8, 11} are also present in {5, 10, 15}:

  • Is 8 in R? No.
  • Is 11 in R? No.

Since neither 8 nor 11 are present in set R, all the elements from (M ∩ P) remain. Therefore, (M ∩ P) - R = {8, 11}.

Clarifying the ni Notation

Now, let's circle back to that ni notation. If ni(X) is meant to represent the cardinality of a set X (which is the number of elements in the set), then the question ni(MP) - R could be interpreted differently. Let's assume MP means M ∩ P.

  1. Calculate M ∩ P: We found this to be {8, 11}.
  2. Calculate ni(M ∩ P): This would be the number of elements in {8, 11}, which is 2. So, ni(M ∩ P) = 2.
  3. The operation becomes 2 - R. However, you cannot subtract a set (R) from a number (2) in standard arithmetic or set theory. This suggests that either the notation ni is specific to a particular context or textbook, or there might be a misunderstanding in how it's applied here.

Possible Interpretation 1: ni(A - B)

If the question meant ni((M ∩ P) - R), then:

  • (M ∩ P) - R = {8, 11} (as calculated above).
  • ni({8, 11}) would be 2.

Possible Interpretation 2: ni(A) - ni(B)

If the question implied finding the difference in the number of elements between two sets, it might have been phrased differently, perhaps asking for ni(M ∩ P) - ni(R). In that case:

  • ni(M ∩ P) = 2.
  • ni(R) = 3 (since R = {5, 10, 15}).
  • ni(M ∩ P) - ni(R) = 2 - 3 = -1. This result (a negative number) is valid if we are just comparing cardinalities, but it doesn't represent a set.

Given the typical structure of such problems, the most probable intent was to find the resulting set of the operation (M ∩ P) - R, and the ni might have been a typo or an extraneous prefix. Therefore, our primary answer, {8, 11}, representing the set difference, is the most likely correct outcome.

Conclusion: Mastering Set Operations

So there you have it, guys! We've successfully navigated the world of set theory, defining our sets P, R, and M, and performing the operations M ∩ P and then (M ∩ P) - R. We found that P = {8, 11, 14}, R = {5, 10, 15}, and M = {4, 5, 8, 9, 11}. The intersection M ∩ P yielded {8, 11}, and finally, the set difference (M ∩ P) - R gave us {8, 11}. Remember, the key to mastering these problems is to take it step-by-step, understand the definitions of the operations (intersection, union, difference), and carefully apply the rules to each element. Don't be afraid to write things down and double-check your work. Keep practicing, and you'll be a set theory superhero in no time! Math is all about logical steps, and once you get the hang of it, it's incredibly rewarding. Keep up the great work!