Question

Draw Venn diagrams for \(A \cup(B \cap C)\) and \((A \cup B) \cap(A \cup C)\). Based on your drawings, do you think \(A \cup(B \cap C)=(A \cup B) \cap(A \cup C) ?\)

Short answer

Yes, the expressions \(A \cup (B \cap C)\) and \((A \cup B) \cap (A \cup C)\) represent the same set regions, thus they are equal.

Step-by-step solution

Draw Venn Diagram for \(A \cup (B \cap C)\)

Draw three circles intersecting each other, each representing sets A, B, and C respectively. The region representing \(B \cap C\) will be the section where circles B and C overlap. The union of this section with set A, denoted as \(A \cup (B \cap C)\), will be the region consisting of the entirety of set A and the overlapping section of B and C.

Draw Venn Diagram for \((A \cup B) \cap (A \cup C)\)

Again, draw three circles intersecting each other for the three sets. The region representing \(A \cup B\) will be the entire region consisting of sets A and B (including their overlap). Similarly, \(A \cup C\) consists of the entire region of sets A and C. The intersection of these two regions, \((A \cup B) \cap(A \cup C)\), will be the sections that are common to both, which is the entirety of set A and the overlapping sections of B and C.

Compare Diagrams

Inspect both diagrams. If the shaded areas of both diagrams coincide completely, the two expressions are equal; if not, they are not equal.

Key concepts

Set Theory

Understanding set theory is crucial when dealing with Venn diagrams. It is a branch of mathematical logic that deals with collections of objects, which we call 'sets'. Each object in a set is called an 'element'.

For example, if we have a set of fruits, oranges and apples could each be elements of this set. A fundamental part of set theory is understanding how different sets can relate to each other. We can have sets that have no elements in common, known as disjoint sets, or sets that share some elements, showing a relationship between them.

When we use Venn diagrams, we are visually representing these relationships. Each circle in a Venn diagram represents a set, and where the circles overlap, we see shared elements. This visual tool is indispensable for exploring complex set relationships easily and intuitively.

Union and Intersection

The concepts of union and intersection are fundamental operations in set theory, and they're visually depicted in Venn diagrams. Let's break them down:

The union of two sets, denoted by the symbol \(\cup\), represents all elements that are in either set, or in both. For instance, if Set A contains apples and Set B contains bananas, then the union \(A \cup B\) would include both apples and bananas.

The intersection of sets, represented by \(\cap\), is the set of elements that are common to all involved sets. Using the previous fruit example, if Set A contains apples and oranges, and Set B contains oranges and grapes, the intersection \(A \cap B\) would only include oranges, the fruit that appears in both sets.

Therefore, when we're looking at Venn diagrams for expressions like \(A \cup (B \cap C)\) and \( (A \cup B) \cap (A \cup C)\), we're seeking regions that represent either the union or the intersection of these sets, which can greatly aid in understanding complex relationships between them.

Mathematical Reasoning

Venn diagrams serve as a tool for mathematical reasoning, which is the process of using logical thinking to solve math problems. In set theory, they enable us to visually reason about relationships between sets.

When you're asked whether \(A \cup (B \cap C) = (A \cup B) \cap (A \cup C)\), you're being prompted to use mathematical reasoning. By drawing the Venn diagrams as described in the solutions, you make abstract set relationships tangible and hence, more comprehensible.

The equality of these two expressions, which does not become apparent through algebra alone, is revealed easily with Venn diagrams. Being able to interpret and compare these graphical representations is a practical application of mathematical reasoning. It combines logical deduction (inferencing the relationship) with spatial visualization (seeing the overlapping areas), proving particularly effective for understanding and proving set identities and equivalences.