Question

Prove \(A \cup B^{\prime}=\left(A^{\prime} \cap B\right)^{\prime}\) by using a Venn diagram.

Short answer

By using a Venn diagram, we have visually demonstrated that the areas represented by both expressions \(A \cup B'\) and \((A' \cap B)'\) are identical. We started by drawing two overlapping circles representing the sets \(A\) and \(B\), found their complements, and then determined the union and intersection of relevant sets. The comparison of the diagrams in Step 7 confirms that the statement \(A \cup B' = (A' \cap B)'\) is true.

Step-by-step solution

Draw a Venn diagram with sets A and B

Start by drawing two overlapping circles inside a rectangle. Label the left circle as set \(A\), the right circle as set \(B\), and the rectangle as the universal set \(U\). The overlapping region in the middle represents the intersection of \(A\) and \(B\), labeled as \(A \cap B\). Step 2: Find B'

Find the complement of set B

The complement of a set, \(B'\), contains all the elements that are NOT part of the set \(B\). In the Venn diagram, this corresponds to every space inside the rectangle (universal set) except the area inside the circle B. Step 3: Find A ∪ B'

Find the union of sets A and B'

The union of two sets, \(A ∪ B'\), contains all the elements that are part of set \(A\) or part of set \(B'\). In the Venn diagram, this is represented by the entire area inside the circle A and the entire area outside the circle B. Step 4: Find A'

Find the complement of set A

The complement of set \(A\), \(A'\), consists of all elements that are NOT part of set \(A\). In the Venn diagram, this corresponds to every space inside the rectangle (universal set) except the area inside the circle A. Step 5: Find A' ∩ B

Find the intersection of sets A' and B

The intersection of two sets, \(A' ∩ B\), contains all elements that are part of both sets, \(A'\) and \(B\). In the Venn diagram, this is represented by the area inside the circle B but outside the circle A. Step 6: Find (A' ∩ B)'

Find the complement of the intersection of sets A' and B

The complement of a set, \((A' ∩ B)'\), contains all the elements that are NOT part of the intersection of sets \(A'\) and \(B\). In the Venn diagram, this corresponds to every space except the area inside the circle B but outside the circle A. (In other words, this is the same area as in Step 3). Step 7: Verify the equality

Verify that A ∪ B' = (A' ∩ B)'

Comparing the result from Step 3 (A ∪ B') and Step 6 ((A' ∩ B)'), it is clear that the areas represented by both expressions in the Venn diagram are identical. Therefore, we have proved the statement \(A ∪ B' = (A' ∩ B)'\) using a Venn diagram.

Key concepts

Set Theory

Set theory is the mathematical study of collections of objects, known as sets. In set theory, a set is defined as a well-defined collection of distinct objects, which can be anything from numbers to letters to symbols. Sets are often denoted by capital letters, such as A, B, or U (for the universal set), and the objects within the set, known as elements, are listed inside curly braces or visually represented in a Venn diagram.

Venn diagrams are powerful tools used to visually illustrate relationships between sets. These diagrams showcase various operations such as unions, intersections, and complements by depicting sets as overlapping circles within a universal set.

The concept of set theory is fundamental to mathematics, as it underpins various other subjects like logic, probability, and statistics. Understanding set theory concepts is essential for pursuing higher mathematical education and solving complex problems.

Complement of a Set

The complement of a set, typically denoted by a prime symbol as in A' or B', refers to all the elements that are not in the specified set A or B, but are in the universal set U. For instance, if the universal set U contains all natural numbers and set A includes the numbers 1, 2, and 3, then the complement of set A, A', consists of all natural numbers except 1, 2, and 3.

In terms of Venn diagrams, the complement is represented by the area in the rectangle (universal set) that is not occupied by the set in consideration. Interestingly, the complement of a set's complement brings you back to the original set, affirming the property that A = (A')'. Understanding set complements is crucial for working with probabilities, logical reasoning, and in areas where excluding specific criteria is important.

Union and Intersection of Sets

The union of sets, denoted by the symbol ∪, represents a set containing all the distinct elements that are in either of the sets. For example, if A = {1, 2, 3} and B = {3, 4, 5}, then the union A ∪ B = {1, 2, 3, 4, 5}. In a Venn diagram, the union is depicted by shading the areas of both sets.

Conversely, the intersection of sets, denoted by the symbol ∩, includes all elements that are common to both sets. Continuing the previous example, the intersection A ∩ B = {3} since 3 is the only number common to both sets A and B. In a Venn diagram, the intersection is indicated by the overlapping area of the sets’ circles.

Understanding union and intersection is fundamental in combining information from different sources or finding commonalities between data sets. These concepts are widely applied in database theory, logic, and are fundamental operations in set theory.