Sets M And N: Union, Intersection, And Difference
Let's dive into a set theory problem! We're given two sets, M and N, defined using inequalities. Our mission, should we choose to accept it (and we do!), is to find their union, intersection, and the difference between N and M. Sounds like fun, right? Let's break it down step-by-step.
Understanding the Sets: Defining M and N
First, we need to get a crystal-clear picture of what these sets actually contain.
Set M: The Realm of -2 to 4
Set M is defined as all real numbers (that's what the 'x ∈ R' means) that are greater than or equal to -2, but strictly less than 4. So, we're talking about numbers like -2, -1, 0, 1, 2, 3, 3.999, and everything in between. But 4 itself is not included, hence the 'less than' sign. We can visualize this on a number line as a closed interval at -2 (because -2 is included) and an open interval at 4 (because 4 is not included). This is crucial for understanding the set's boundaries and how it interacts with other sets.
Set N: Absolute Value Adventures
Set N is a bit trickier because it involves an absolute value inequality: |2x+1| ≤ 3. Remember, absolute value means the distance from zero. So, this inequality is saying that the distance between 2x+1 and zero is less than or equal to 3. To solve this, we need to split it into two separate inequalities:
- 2x + 1 ≤ 3
- -(2x + 1) ≤ 3
Let's solve the first one: 2x + 1 ≤ 3. Subtracting 1 from both sides gives us 2x ≤ 2. Dividing both sides by 2, we get x ≤ 1. So, all numbers less than or equal to 1 are potential members of set N.
Now, let's tackle the second inequality: -(2x + 1) ≤ 3. Distributing the negative sign, we get -2x - 1 ≤ 3. Adding 1 to both sides gives us -2x ≤ 4. Now, here's a crucial step: when we divide both sides by a negative number, we need to flip the inequality sign. So, dividing by -2, we get x ≥ -2. So, all numbers greater than or equal to -2 are also potential members of set N.
Combining these two results, we find that set N includes all real numbers between -2 and 1, inclusive. That means -2, 1, and everything in between belongs to N. This forms a closed interval on the number line, from -2 to 1.
Operations on Sets: Unleashing the Power of Union, Intersection, and Difference
Now that we've decoded what sets M and N are all about, let's get our hands dirty with the set operations: union, intersection, and difference. These operations allow us to combine and compare sets in meaningful ways, revealing relationships and creating new sets.
a) M ∪ N: The Union - A Grand Gathering
The union of two sets, denoted by M ∪ N, is like inviting everyone from both sets to a party. It's the set containing all elements that are in M, or in N, or in both. No one is left out! The union essentially combines the ranges of values covered by both sets, giving us a larger, more inclusive set. So, M ∪ N includes all numbers that are in M or N or both. To find M ∪ N, we need to consider the intervals defined by M and N and see where they overlap and extend.
Set M covers the range [-2, 4), meaning it includes all numbers from -2 (inclusive) up to, but not including, 4. Set N covers the range [-2, 1], including all numbers from -2 to 1. When we combine these, we start at the lowest value, which is -2, and extend to the highest value covered by either set. In this case, that's 4 (but not including 4, because 4 is not in M and therefore not in the union). So, M ∪ N will cover the interval from -2 up to 4. Essentially, we're merging the two intervals, taking care not to duplicate any elements. The union operation is fundamental in many areas of mathematics and computer science, allowing us to combine different sets of data or conditions into a single, comprehensive set. Think of it as merging two lists into one, without losing any information.
Therefore, M ∪ N = {x ∈ R | -2 ≤ x < 4}.
b) M ∩ N: The Intersection - Where Worlds Collide
The intersection of two sets, denoted by M ∩ N, is a bit more exclusive. It's like a secret club where only members who belong to both M and N are allowed. It's the set containing only the elements that are common to both sets. The intersection gives us the overlap between the two sets, showing us which elements are present in both. Finding M ∩ N means identifying the range of values that are present in both set M and set N. This operation is crucial for identifying shared properties or conditions in various scenarios.
Looking at our sets, M covers [-2, 4) and N covers [-2, 1]. The overlap occurs between -2 and 1. Both sets include -2, and both sets include all numbers up to 1. However, set M extends beyond 1, up to 4, but those numbers are not in N. Therefore, the intersection is the range of numbers from -2 to 1, inclusive. The intersection is a powerful tool in many fields, from database queries (finding customers who meet multiple criteria) to logic (identifying conditions that must be simultaneously true).
Thus, M ∩ N = {x ∈ R | -2 ≤ x ≤ 1}.
c) N - M: The Difference - What N Has That M Doesn't
The difference between two sets, denoted by N - M, is a bit like taking away everything that N and M have in common, leaving only what's unique to N. It's the set containing elements that are in N but not in M. We're essentially subtracting the elements of M from N. So, N - M includes all numbers that are part of set N but are not part of set M. To find N - M, we need to carefully consider the ranges covered by N and M and identify what's left in N after removing the elements it shares with M. This operation is useful for isolating specific data or conditions that are unique to one set compared to another.
Set N is [-2, 1], and set M is [-2, 4). The overlap (the intersection we just calculated) is [-2, 1]. So, if we take away the intersection from N, we're left with nothing. This is because N is entirely contained within M up to the value of 1. Every element in N is also in M, so after subtracting M from N, we have an empty set. The difference operation is used in various applications, such as data filtering (removing duplicates) and logical reasoning (identifying conditions that are true in one case but not another).
Therefore, N - M = ∅ (the empty set).
Wrapping Up: Mastering Set Operations
So, there you have it, guys! We've successfully navigated the world of sets, calculating the union, intersection, and difference between sets M and N. By understanding the definitions of these sets and the fundamental set operations, we've cracked the code and arrived at our solutions. Set theory might seem abstract, but it's a powerful tool in mathematics and computer science, helping us organize, compare, and manipulate collections of objects. Keep practicing, and you'll be a set theory pro in no time!
In summary:
- M ∪ N = {x ∈ R | -2 ≤ x < 4}
- M ∩ N = {x ∈ R | -2 ≤ x ≤ 1}
- N - M = ∅
Keep exploring the fascinating world of math, and remember, every problem is just a puzzle waiting to be solved!