Given the sets S1={2,4,6}, S2={7,2,6}, S3={4,2,6}, and...

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) $S_{3}=\{1,2\}$

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) $S_{3}=\{1,2\}$

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Key Concepts

Finite vs Infinite Sets

Finite sets have a countable number of elements, whereas infinite sets, such as the set of real numbers, have no bound on the number of elements. This distinction is important when discussing properties and operations on sets.

Empty Set as a Universal Subset

The empty set is considered a subset of every set because it has no elements that could possibly violate the subset condition. This universality simplifies many aspects of set theory.

Set Equality

Set equality requires that two sets have exactly the same members regardless of order or repetition. This concept is central in determining if two sets can be considered identical.

Set Membership

Set membership is the concept that determines whether an element is part of a set, indicated by the symbol ?. It is essential for assessing the inclusion or exclusion of individual elements within a set.

Subset and Superset Relations

A subset is a set all of whose elements are also contained within another set (denoted by ?), while a superset contains all elements of another set (denoted by ?). These relations are foundational when comparing sets in terms of their elements.

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