Problem 7 - Applied Discrete Structures

Question

Let \(A=\\{+,-\\}\) and \(B=\\{00,01,10,11\\}\). (a) List the elements of \(A \times B\) (b) How many elements do \(A^{4}\) and \((A \times B)^{3}\) have?

Step-by-Step Solution

Answer

(a) \{(+,00), (+,01), (+,10), (+,11), (-,00), (-,01), (-,10), (-,11)\}. (b) 16 elements in \(A^4\), 512 elements in \((A \times B)^3\).

Step 1: Understanding Cartesian Product

The Cartesian product of two sets, say \(A\) and \(B\), denoted as \(A \times B\), is the set of all ordered pairs \((a, b)\) where \(a \in A\) and \(b \in B\). Here, \(A = \{+, -\}\) and \(B = \{00, 01, 10, 11\}\). Each element of \(A\) will pair with each element of \(B\).

Step 2: List Elements of A × B

Calculate \(A \times B\) by creating pairs: \((+, 00)\), \((+, 01)\), \((+, 10)\), \((+, 11)\), \((- , 00)\), \((- , 01)\), \((- , 10)\), \((- , 11)\). Thus, \(A \times B = \{ (+, 00), (+, 01), (+, 10), (+, 11), (-, 00), (-, 01), (-, 10), (-, 11) \} \).

Step 3: Calculate Elements of A^4

\(A^4\) refers to the cartesian product of the set \(A\) with itself 4 times: \(A \times A \times A \times A\). Since \(A\) has 2 elements, \(A^4\) will have \(2^4 = 16\) elements.

Step 4: Calculate Elements of (A × B)^3

\((A \times B)^3\) refers to the cartesian product of the set \(A \times B\) with itself 3 times. Since \(A \times B\) has 8 elements (as shown in step 2), \((A \times B)^3\) will have \(8^3 = 512\) elements.

Key Concepts

Understanding Ordered Pairs

An ordered pair is a fundamental concept in mathematics that involves two elements arranged in a specific sequence. This is generally denoted by \((a, b)\), where \(a\) is the first element and \(b\) is the second. Unlike regular pairing, the order in which these elements appear is crucial.
In set theory, when we talk about sets used in ordered pairs, sets themselves do not have a specific order. However, when we create an ordered pair from two sets \(A\) and \(B\), the first element is always from \(A\) and the second from \(B\).

• For example: If \(A = \{+, -\}\) and \(B = \{00, 01, 10, 11\}\), the ordered pair \((+, 00)\) is different from \((00, +)\). They are distinct because the order of elements matters.
• The Cartesian product \(A \times B\) results in all possible ordered pair combinations from sets \(A\) and \(B\).

Ordered pairs are foundational in coordinate systems, allowing us to represent points on a plane as \((x, y)\). It's common in disciplines like computer science for operations involving data structures and functions.

Exploring Set Theory

Set theory is the study of collections of objects, called sets. It is one of the most fundamental areas in mathematics and forms the core of various topics. A set is simply a collection of distinct objects, which can be anything: numbers, symbols, or even other sets.

• In this context, the sets \(A = \{+, -\}\) and \(B = \{00, 01, 10, 11\}\) are considered.
• Each set must contain unique elements: no duplications are allowed. This distinct nature makes operations, such as union and intersections, useful for defining relationships between different sets.

Set theory introduces the idea of the "universal set," which encompasses all possible elements under consideration. Subsets, formed from a universal set, can then be examined individually or in combination.
The Cartesian product \(A \times B\) involving set theory leads us to the idea that we can map elements from one set to another, creating a new set of ordered pairs.

Basics of Combinatorics

Combinatorics deals with the counting, arrangement, and combination of elements within sets. It examines how we can put together various components in different ways. The fundamental principle in combinatorics is counting the number of possible outcomes.

• The total count is achieved using basic multiplication principles. For example, if there are \(n\) ways to do one thing and \(m\) ways to do another, there are \(n \times m\) ways to do both.
• In the exercise, \(A^4\) means four elements picked from \(A\), which gives us \(2^4 = 16\) since there are two choices (\(+, -\)) for each of the four positions.
• For \((A \times B)^3\), each "](A \times B)" set has 8 elements, and since it is cubed, we end up with \(8^3 = 512\) possible combinations of ordered triplets.

Combinatorics provides the tools necessary to efficiently handle large volumes of permutations and combinations, which are vital in fields like computer science, probability, and statistics.