Question

Draw a Venn diagram for \(\bar{A},\) where \(A\) is a subset of a universal set \(U\).

Short answer

The Venn diagram for \(\bar{A}\), where \(A\) is a subset, of the universal set \(U\) will be a rectangle representing the universal set \(U\), a circle inside the rectangle representing subset \(A\), and the shaded area outside the circle but inside the rectangle representing the complement of the subset \(A\), \(\bar{A}\).

Step-by-step solution

Identifying the Universal Set

Firstly, determine the universal set \(U\). It represents all possible members related to our object of study, and contains all the elements including set \(A\) and its complement in our context. In our Venn diagram, we depict this by drawing a rectangle.

Identifying the Set A

Next, identify set \(A\). It is said to be a subset of set \(U\), which means all elements of \(A\) are also elements of \(U\). We represent \(A\) inside the rectangle (which is \(U\)) by drawing a circle.

Identifying Complement of Set A

Then, consider the complement of set \(A\) denoted by \(\bar{A}\). In the terms of the universal set, \(\bar{A}\) includes all the elements of the universal set \(U\) that are not in set \(A\). Hence, this is depicted in the Venn diagram by shading or coloring the area in the rectangle, excluding the circle which is set \(A\).

Key concepts

Complement of a Set

Imagine you have a bag of colored marbles. If we define set A as containing just the red marbles, then the complement of set A, denoted as \( \bar{A} \), is like taking all the marbles that are not red out of the bag. In mathematical terms, the complement of set A refers to elements that are in the universal set U but not in set A. When we draw a Venn diagram, the complement of A is represented by shading the area outside the circle representing set A but within the boundaries of the rectangle that represents the universal set. This helps us visualize the concept of 'everything else but A'.
To summarize, the complement of set A consists of:

• Elements in the universal set U.
• Excluding the elements that are in set A.

It is important to note that in a Venn diagram, the complement is usually depicted by shading the area outside the designated set A while still being within the confines of the universal set.

Universal Set

The term 'universal set' may sound grand, and in a way, it is. It represents the entire collection of all possible elements under consideration. Think of it as the 'stage' upon which all our mathematical 'actors' (the elements) perform. In the context of sets, the universal set U includes every single element that can be discussed within a given context. When we represent this in a Venn diagram, we draw a rectangle to symbolize this all-encompassing set.
Key points about the universal set:

• Inclusive of all elements pertinent to the discussion.
• Depicted by a rectangle in Venn diagrams.
• Acts as the reference set for other subsets and complements within the diagram.

Whenever you encounter problems involving sets, remember that the universal set is your reference point, the 'whole picture' to which other sets are compared.

Subset

The term 'subset' might make you think of a piece of a puzzle fitting snugly within the larger picture. A subset is exactly that — a set where every element also belongs to another, larger set, known as the universal or parent set. In symbols, if set A is a subset of set U (the universal set), we write this as \( A \subseteq U \).
To understand this with a real-world analogy, imagine you have a cookie jar. This cookie jar (the universal set) contains chocolate chip cookies, peanut butter cookies, and oatmeal cookies. If we take out just the chocolate chip cookies and place them on a plate, that plate of cookies (set A) is a subset of the cookies in the jar because every chocolate chip cookie on the plate is also in the jar.
Here are the key aspects of a subset:

• Every element in a subset is also in the universal set.
• In Venn diagrams, subsets are usually represented by circles inside the rectangle representing the universal set.
• Not every subset is a proper subset; a set is considered a proper subset if it does not contain all the elements of the universal set.