Problem 47 - Discrete Mathematics with Applications

Question

Mark each as true or false, where \(A\) and \(B\) are arbitrary finite languages. \(|A B|=|B A|\)

Step-by-Step Solution

Answer

True. For arbitrary finite languages A and B, the number of elements in the concatenation of A and B (\(A B\)) is equal to the number of elements in the concatenation of B and A (\(B A\)). If A has m elements and B has n elements, both concatenations will have \(m * n\) elements, since multiplication is commutative.

Step 1: Understanding Concatenation

To solve this problem, we must first understand the concept of concatenation. The concatenation of two languages A and B, denoted as \(A B\), results in a new language formed by each word from A followed by each word from B. Similarly, the concatenation of B and A, denoted as \(B A\), is formed by each word from B followed by each word from A. Step 2: Example Concatenation

Step 2: Evaluate an Example

Let's consider an example with two finite languages to better understand how concatenation works: Let A = \(\{x, y\}\) and B = \(\{1, 2\}\) Now, we can find \(A B\) and \(B A\): \(A B = \{x1, x2, y1, y2\}\) \(B A = \{1x, 1y, 2x, 2y\}\) Notice that the number of elements in both sets are equal. Step 3: General Formulation

Step 3: Formulate a General Rule

We need to find a general rule that will hold for any arbitrary finite languages A and B. Let's use the following definitions for A and B: Let A = \(\{a_1, a_2, ..., a_m\}\) Let B = \(\{b_1, b_2, ..., b_n\}\) To form the concatenation of A and B, we need to concatenate each element of A with each element of B. Since there are m elements in A and n elements in B, there will be \(m*n\) elements in their concatenation. Step 4: Evaluate the Equality

Step 4: Check the Number of Elements in Both Concatenations

Now, we can check the number of elements in both \(A B\) and \(B A\). For \(A B\), there are \(m * n\) elements, as each element of A is concatenated with each element of B. For \(B A\), there are \(n * m\) elements, as each element of B is concatenated with each element of A. These two quantities are equal, as multiplication is commutative: $$m * n = n * m$$ Step 5: Conclusion

Step 5: Determine if the Statement is True or False

Based on the analysis above, we can conclude that the given statement is true: $$\left|A B\right| =\left|B A\right|$$

Key Concepts

Finite Languages

A finite language is a collection of strings over a given alphabet where the number of strings is limited or finite. In other words, a finite language can be listed or enumerated because it has a definite number of elements. An example of a finite language could be the set of all strings of lowercase English letters that form three-letter words. Since the number of all possible three-letter words is limited, this is a finite language.

When dealing with finite languages in the context of operations like concatenation, it’s essential to understand that even though the languages are finite, the operation can lead to a significant increase in the number of elements. However, the result stays finite as it's the product of finite numbers, which is of particular interest when talking about the cardinality of these sets.

In our exercise, languages 'A' and 'B' were both finite, with elements that can be represented as \(A = \(a_1, a_2, ..., a_m\)\) and \(B = \(b_1, b_2, ..., b_n\)\), where \(m\) and \(n\) are finite numbers representing the cardinality of sets 'A' and 'B', respectively.

Commutative Property

The commutative property is a fundamental principle in basic algebra that applies to many operations, including addition and multiplication of numbers. This property essentially states that changing the order of the numbers involved in the operation does not change the result. In mathematical terms, for any two elements \(a\) and \(b\), the property is defined as \(a + b = b + a\) for addition and \(a * b = b * a\) for multiplication.

In the context of our exercise, we apply the commutative property to the operation of concatenation for finite languages. Even though natural language concatenation (such as sentences in English) does not generally obey the commutative property because 'cat dog' is not the same as 'dog cat', in the language concatenation under discussion, we are only concerned with the cardinality—or number—of possible concatenations.

With finite languages, as seen in step 4 of the original solution, the cardinality after performing concatenation operations is commutative, meaning the number of elements obtained from \(A B\) and \(B A\) will be the same, because \(m * n = n * m\). This shows how mathematical properties can sometimes manifest in unexpected ways within theoretical computer science concepts.

Cardinality of Sets

The term 'cardinality' refers to the number of elements in a set. When sets are finite, their cardinality is simply the count of distinct elements. For example, the cardinality of the set \(\{a, b, c\}\) is 3. In more formal terms, if the set 'S' has 'n' number of elements, its cardinality is denoted by \(|S| = n\).

Understanding cardinality becomes particularly important in computational theory, where we are often concerned with the size of sets, such as the number of possible outputs from an algorithm or the number of words in a language. In operations like concatenation, we use the cardinality of each set to determine the cardinality of the resulting set.

In the concatenation of two finite languages 'A' and 'B', the cardinality of the resultant set \(A B\) is the product of the cardinalities of 'A' and 'B'. This is because every element of 'A' is paired with every element of 'B', thus if \(A\) has \(m\) elements and \(B\) has \(n\) elements, then the concatenated set \(A B\) has \(m * n\) elements, as illustrated in the example from our step-by-step solution. This multiplication of cardinalities again reflects the commutative property, because no matter the order of concatenation, the resulting cardinality remains the same.