Problem 13 - Discrete Mathematics with Applications

Question

Using the sets \(A=\\{a, b, e, h\\}, B=\\{b, c, e, f, h |, C=\\{c, d, f, g\\}, \text { and }\) \(U=\\{a, \ldots, h\\},\) find the binary representation of each set. $$ B^{\prime} $$

Step-by-Step Solution

Answer

The binary representation of the complement set \(B'\) is \(10100100\).

Step 1: Identify the universal set and the given set

The universal set U is given as \(U = \{a, b, c, d, e, f, g, h\}\) and the set B as \(B = \{b, c, e, f, h\}\).

Step 2: Find the complement of set B

To find the complement of set B (denoted as \(B'\)), we need to identify the elements in the universal set U that are not in set B. So, we have \(B' = U - B = \{a, d, g\}\).

Step 3: Convert the complement set into binary representation

We need to represent the elements of \(B'\) in binary form, and the order of the elements should follow the U's order. We'll assign '1' if the element is present in \(B'\) and '0' if it's not present. Here's the binary representation: - a is present, so '1' - b is not present, so '0' - c is not present, so '0' - d is present, so '1' - e is not present, so '0' - f is not present, so '0' - g is present, so '1' - h is not present, so '0' So, the binary representation of \(B'\) is \(10100100\).

Key Concepts

Complement of a Set

Imagine you have a pizza with toppings you love, except for a few you'd rather not have on your slice. In set theory, the toppings you don't want can be thought of as the complement of a set. To grasp this concept better, consider a universal set, akin to an entire selection of pizza toppings. The complement of a set includes all the elements from this universal spread that are not in the original set.

For instance, if set A has pepperoni and mushrooms, and the universal set has sausage, onions, pepperoni, and mushrooms, the complement of A would contain sausage and onions—since those are the elements outside of set A, but within the universal options. Mathematically, if set A is a subset of a universal set U, then the complement of A, denoted as A', is given by all the elements in U that are not in A.

Applying the concept to our pizza-topping example—A' would be the toppings A does not include. Always keep in mind that the elements of the complement are part of the universal set and are determined by what is absent in the original set.

Universal Set

In the world of sets, the universal set, denoted often as U, is the container that holds all the possible elements under discussion. You can think of it as a box that has every imaginable piece of LEGO you might use to build something. Everything outside this box does not exist for your current project.

Every other set in your discussion is a subset of this universal set. This means that any element found in a subset must also be an element of the universal set. The universal set concept is crucial because it establishes the domain for all other subsets we talk about and defines what the complements are made up of, as each element in the complement must be in the universal set but not in the set we're evaluating.

Remember, the definition of the universal set is context-dependent; it changes based on what you're discussing. If you're dealing with letters of the alphabet, U might consist of all 26 letters. But if you're working with numbers, U could be all integers, reals, or another appropriate set.

Element of a Set

Whenever you see a set, think of it as a collection of distinct objects, just like a classroom filled with students. Each student in that room is an element of the set of students. Similarly, in set theory, an element is an individual object that belongs to a set.

An element can be anything—numbers, letters, symbols, points in space, and so on. To denote that an element, let's say 'e', belongs to a set A, we use the symbol ∈ and write it as 'e ∈ A'. If element x does not belong to a set A, we write 'x ∉ A'. It's essential to recognize each element's unique presence in a set because this forms the foundation for understanding operations like unions, intersections, and complements.

For instance, if you have a box of crayons, each crayon is an element of the set. If you're looking for the red crayon and you find it, you've essentially located an element of the set. Acknowledging an item as an element of a set is the first step in performing any operation or analysis within set theory.