Question

Given the sets \(S_{1}=\\{2,4,6\\}, S_{2}=\\{7,2,6\\}, S_{3}=\\{4,2,6\\},\) and \(S_{4}=\\{2,4\\},\) which of the following statements are true? (a) \(S_{1}=S_{3}\) (b) \(S_{1}=R\) (set of real numbers) (c) \(8 \in S_{2}\) \((d) 3 \notin S_{2}\) \((e) 4 \notin S_{3}\) \((f) S_{4} \subset R\) \((g) S_{1} \supset S_{4}\) \((h) \varnothing \subset S_{2}\) (i) \(S_{3}=\\{1,2\\}\)

Short answer

Statements (a), (d), (f), (g), and (h) are true.

Step-by-step solution

Evaluate (a)

Compare the sets \(S_1=\{2,4,6\}\) and \(S_3=\{4,2,6\}\). Since sets are equal if they contain exactly the same elements, regardless of order, we see both sets \(S_1\) and \(S_3\) contain the elements 2, 4, and 6. Thus, \(S_1 = S_3\) is true.

Evaluate (b)

Check if \(S_{1}=R\). The set \(S_1\) only contains the elements \(\{2, 4, 6\}\), whereas the set of real numbers \(R\) is infinitely large and includes all possible real numbers. Therefore, \(S_{1} eq R\) is false.

Evaluate (c)

Check membership of 8 in \(S_2=\{7, 2, 6\}\). The number 8 is not listed as an element of \(S_2\), thus \(8 \in S_2\) is false.

Evaluate (d)

Check non-membership of 3 in \(S_2=\{7, 2, 6\}\). The number 3 is not in \(S_2\), so \(3 otin S_{2}\) is true.

Evaluate (e)

Check non-membership of 4 in \(S_3=\{4, 2, 6\}\). The number 4 is indeed present in \(S_3\), so \(4 otin S_3\) is false.

Evaluate (f)

Check if \(S_4=\{2, 4\} \subset R\). Since every element in \(S_4\) is a real number, \(S_4\) is indeed a subset of \(R\). Thus, \(S_4 \subset R\) is true.

Evaluate (g)

Check if \(S_1=\{2, 4, 6\}\) is a superset of \(S_4=\{2, 4\}\). Since all elements of \(S_4\) are contained in \(S_1\), \(S_1 \supset S_4\) is true.

Evaluate (h)

Check if the empty set \(\varnothing\) is a subset of \(S_2=\{7, 2, 6\}\). By definition, the empty set is a subset of every set. Therefore, \(\varnothing \subset S_2\) is true.

Evaluate (i)

Compare \(S_3=\{4, 2, 6\}\) to \(\{1, 2\}\). They contain different elements, so \(S_3=\{1, 2\}\) is false.

Key concepts

Subset

A subset is a fundamental concept in set theory, essential for understanding how different sets relate to one another. If every element of a set \( A \) is also an element of a set \( B \), we say that \( A \) is a subset of \( B \), which is denoted as \( A \subseteq B \). For example, in the exercise above, it is stated that \( S_4 = \{2, 4\} \subset R \), meaning every element in \( S_4 \) is found in the set of all real numbers \( R \). It's important to remember that the subset relation is inclusive, so \( A \) could be equal to \( B \). However, if there is at least one element in \( B \) that is not in \( A \), then \( A \) is a "proper" subset of \( B \), written as \( A \subset B \). Moving back to the exercise, the statement \( S_1 \subset S_3 \) would typically be false if \( S_1 \) and \( S_3 \) are equal because subsets must specifically be smaller in the "proper" context.

Superset

Just as a subset is contained within another set, a superset contains another set. When set \( A \) holds all the elements of set \( B \), we call \( A \) a superset of \( B \), written as \( A \supseteq B \). An example from the exercise is \( S_1 \supset S_4 \), where \( S_1 \), containing all elements from \( S_4 \) and more, qualifies as a superset. It's crucial to recognize that the notation \( \supseteq \) includes the possibility that \( A \) and \( B \) are equal. This inclusive definition mirrors that of subsets. If \( A \) strictly has more elements than \( B \), this would be called a "proper" superset, denoted by \( A \supset B \). Recognizing the symbiotic relationships between subsets and supersets helps solidify how sets function in relation to each other.

Empty Set

The empty set, often denoted by \( \varnothing \) or \( \{\} \), is a unique set that contains no elements. It plays a special role in set theory because it is universally a subset of any set, making it an inherent building block of mathematical reasoning. In the exercise, we see the statement \( \varnothing \subset S_2 \). By definition, this is true because the empty set doesn't require any prerequisites to be considered a subset; it vacantly fits into any other set by default. This property of the empty set simplifies many logical arguments and proofs within mathematics as it ensures foundational stability, serving as an anchor point for more complex constructions.

Set Equality

Set equality is a straightforward but vital concept in understanding set relations. Two sets \( A \) and \( B \) are considered equal if they consist of exactly the same elements, meaning \( A = B \). The order doesn’t matter in sets, so long as every member of \( A \) is in \( B \) and vice-versa. For example, in the exercise, \( S_1 = \{2, 4, 6\} \) and \( S_3 = \{4, 2, 6\} \) are evaluated. Since both hold the same elements regardless of order, they are equal, confirming \( S_1 = S_3 \) as true. This principle is fundamental: being equal means having no element in one that isn’t in the other. Understanding set equality is pivotal in solving many mathematical queries that rely on precise matching of sets.