Problem 11 - Discrete Mathematics with Applications

Question

Let \(A\) and \(B\) be two sets such that \(|A|=2 a-b,|B|=2 a,|A \cap B|=a-b\) and \(|U|=3 a+2 b .\) Find the cardinality of each set. Find \(|A \cap B|\) if \(|A|=2 a,|B|=a,\) and \(|A \cup B|=2 a+b.\)

Step-by-Step Solution

Answer

In the first scenario, the cardinalities of the sets are \(|A| = 2a - b\), \(|B| = 2a\), and \(|A ∩ B| = \frac{a}{2}\). In the second scenario, \(|A| = 2a\), \(|B| = a\), and \(|A ∩ B| = a - b\).

Step 1: Identify the given information

The exercise provides us with the following information regarding the sets A, B, and their intersection: \[|A| = 2a - b\] \[|B| = 2a\] \[|A \cap B| = a - b\] \[|U| = 3a + 2b\] Our ultimate goal is to find the cardinality of each set and their intersection.

Step 2: Analyzing the universal set

Since |U| = 3a + 2b, we can analyze the whole universal set to find the cardinality of the two sets A and B. We know that the whole Universal set can be split into three disjoint sets: |A| except |A ∩ B|, |B| except |A ∩ B|, and |A ∩ B|. Therefore, \[|U| = (|A| - |A \cap B|) + (|B| - |A \cap B|) + |A \cap B|\] We have the values of |U|, |A|, |B|, and |A ∩ B|. Replace these values in the equation: \[3a + 2b = (2a - b - a + b) + (2a - a + b) + (a - b)\]

Step 3: Solve for a and b

Now, simplify and solve for a and b in the equation: \[3a + 2b = a + 2a + a\] \[3a + 2b = 4a\] Now, subtract 3a from both sides to find the value of b: \[2b = a\] Next, substitute the value of a back into the equation |A ∩ B| = a - b: \[|A \cap B| = a - \frac{a}{2} = \frac{a}{2}\]

Step 4: Solve for |A ∩ B| in the second case

The exercise also asks us to find |A ∩ B| given the following values: \[|A| = 2a\] \[|B| = a\] \[|A \cup B| = 2a + b\] Using the same formula, we get: \[|A \cup B| = |A| + |B| - |A ∩ B|\] Substitute the given values: \[(2a + b) = (2a) + a - |A ∩ B|\] Now, solve for |A ∩ B|: \[|A ∩ B| = a - b\] In conclusion, for the first given scenario, the cardinalities of the sets are |A| = 2a - b, |B| = 2a, and |A ∩ B| = a/2. In the second scenario, |A| = 2a, |B| = a, and |A ∩ B| = a - b.

Key Concepts

Set Theory

When delving into the world of mathematics, one can’t help but encounter the fundamental concept of set theory. It's the branch of mathematical logic that studies collections of objects, referred to as sets, and their properties. Think of a set as a gathering of specific, well-defined objects, which could be anything from numbers to letters, or even other sets.

In set theory, the cardinality of a set is a measure of the 'number' of elements of the set. For example, the set \(A = \(1, 2, 3\)\) has a cardinality of 3, denoted as \(│A│ = 3\). This concept becomes particularly useful when comparing the sizes of sets, determining if they are finite, infinite, and whether they have the same cardinality or size.

A crucial aspect to remember is that in set theory, the way elements are arranged or repeated does not affect the set. For instance, \( \(1, 2, 3\) \) and \( \(3, 2, 1\) \) are considered the same set because they consist of the same elements. Likewise, the set \( \(1, 1, 2, 3\) \) is still treated as \( \(1, 2, 3\) \) because repetitions are ignored.

Intersection of Sets

The intersection of sets is a concept in set theory used to describe elements that two or more sets have in common. Symbolically, the intersection of two sets A and B is denoted as \(A \cap B\), and it consists of all the elements that are in both A and B.

To illustrate, suppose we have a set \(A = \{1, 2, 3\}\) and another set \(B = \{2, 3, 4\}\). The intersection \(A \cap B\) would be \(\{2, 3\}\) because these are the numbers that appear in both sets. Here is a simple but powerful rule to remember: if there are no common elements, the intersection of those sets is the empty set, symbolically represented as \(\varnothing\) or \(\{\}\).

Applying Intersection to the Exercise

When tasked to solve problems related to set intersections, such as finding \(│A \cap B│\), you focus on the shared elements within the given sets. In our exercise, we used the cardinalities of A and B and other information to deduce the size of \(│A \cap B│\), which is a critical step in solving many problems involving sets.

Universal Set

In set theory, the universal set, often denoted by \( U \), is the set that contains all objects or elements considered for a particular problem or situation - essentially, it’s the ‘set of all things.' Any set within a particular discussion is a subset of this universal set.

For example, if we're dealing with natural numbers, the universal set could be all natural numbers. Within that universal set, you can have numerous subsets, such as even numbers, prime numbers, etc. The understanding of a universal set provides a comprehensive backdrop against which other sets are considered.

Universal Set in Our Exercise

In the exercise provided, the universal set U encompasses all the elements in sets A and B, including those not shared (not in \(A \cap B\)). To find the cardinality of the universal set, we combined the unique elements of A and B with their intersection. This is pivotal because it prevents double-counting the shared elements when calculating the total number of distinct elements within the universal set.