Problem 5 - Applied Discrete Structures

Question

Let \(A=\\{0,2,3\\}, B=\\{2,3\\},\) and \(C=\\{1,5,9\\} .\) Determine which of the following statements are true. Give reasons for your answers. (a) \(3 \in A\) (e) \(A \subseteq B\) (b) \\{3\\}\(\in A\) (f) \(\emptyset \subseteq C\) (c) \\{3\\}\(\subseteq A\) (g) \(\emptyset \in A\) (d) \(B \subseteq A\) (h) \(A \subseteq A\)

Step-by-Step Solution

Answer

True: (a), (c), (d), (f), (h); False: (b), (e), (g).

Step 1: Evaluate statement (a)

Statement (a) is \(3 \in A\). This asks if the element \(3\) is in the set \(A\). Since \(A = \{0,2,3\}\) and \(3\) is listed as one of the elements, this statement is true.

Step 2: Evaluate statement (b)

Statement (b) is \(\{3\} \in A\). This asks if the set containing the single element \(3\) is an element of \(A\). Since \(A\) only contains numbers, not sets, this statement is false.

Step 3: Evaluate statement (c)

Statement (c) is \(\{3\} \subseteq A\). This asks if the set containing \(3\) is a subset of \(A\). A set is a subset of another if all its elements are contained in the other. Since \(3\) is in \(A\), this statement is true.

Step 4: Evaluate statement (d)

Statement (d) is \(B \subseteq A\). This checks if all elements of \(B\) are in \(A\). Since \(B = \{2,3\}\) and both \(2\) and \(3\) are in \(A\), this statement is true.

Step 5: Evaluate statement (e)

Statement (e) is \(A \subseteq B\). This checks if every element of \(A\) is in \(B\). Since \(A = \{0,2,3\}\) and \(0\) is not in \(B\), this statement is false.

Step 6: Evaluate statement (f)

Statement (f) is \(\emptyset \subseteq C\). The empty set is a subset of every set because it contains no elements to conflict with any other set, so this statement is true.

Step 7: Evaluate statement (g)

Statement (g) is \(\emptyset \in A\). This asks if the empty set is an element of \(A\). Since \(A\) only contains numbers, this statement is false.

Step 8: Evaluate statement (h)

Statement (h) is \(A \subseteq A\). A set is always a subset of itself, so this statement is true.

Key Concepts

Subsets

In set theory, the concept of subsets is very important. A subset is a set whose elements are all contained within another set. If every element of set B is also an element of set A, then B is a subset of A.

• The symbol for subset is \( \subseteq \).
• For example, if \( B = \{2, 3\} \) and \( A = \{0, 2, 3\} \), since both elements 2 and 3 are in set A, \( B \subseteq A \) is true.
• The statement \( A \subseteq A \) is always true because any set is automatically a subset of itself.
• The empty set \(\emptyset\) is also a subset of every set because there are no elements to violate the condition of being a subset.

Understanding the idea of subsets helps in comparing different sets and analyzing their relationships.

Elements of a Set

When discussing sets, it's crucial to understand what constitutes an element of a set. Simply put, an element is a single object or number contained within a set.

• The symbol \( \in \) denotes that a particular element is a member of a set.
• For instance, with the set \( A = \{0, 2, 3\} \), the notation \( 3 \in A \) is used to indicate that 3 is an element of set A.
• Conversely, \( \{3\} \in A \) would be false because \( A \) contains numbers as elements, not the set containing 3 itself.

Recognizing elements of a set is fundamental for determining membership in a set and for performing operations with sets.

Empty Set

The empty set, denoted by \( \emptyset \), is a unique set in mathematics. This set has no elements, making it the basis for many concepts in set theory.

• Despite having no elements, \( \emptyset \) is considered a subset of every set. This is because there are no elements in \( \emptyset \) that could conflict with elements in other sets.
• However, \( \emptyset \in A \) is false, since \( \emptyset \) as an object is not a number or member within set A, but rather a set itself.

Understanding the properties of the empty set is crucial for grasping more complex set concepts and operations. It acts as a starting point for defining and comparing other sets.

Set Membership

Set membership is about determining whether an object belongs to a particular set.

• To express that an element \( x \) belongs to a set \( A \), the notation \( x \in A \) is used. If \( x \) does not belong to \( A \), it is expressed as \( x otin A \).
• In the example exercise, 3 is a member of set A, written as \( 3 \in A \), which is true. This notation is fundamental to identifying relationships between elements and sets.
• Another example is \( \emptyset \in A \). This checks if the empty set is a member of \( A \). In our case, \( A \) only has numbers, making this statement false.

By understanding set membership, you can better interpret whether individual elements or even whole other sets relate to a given set. This forms the basis for more advanced studies in set theory and logical mathematics.