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Set notation is the standardized way of writing down a collection of distinct objects, which are considered the 'elements' of the set. This can be anything from numbers, characters, or even other sets. For instance, the set of all vowels in the English alphabet can be described using set notation as \( \{a, e, i, o, u\} \.

When we want to indicate that a set doesn't contain any elements at all, we use a specific set notation that looks like this: \( \{\} \). This represents an empty set, a fundamental concept in various branches of mathematics, including set theory and probability. An empty set is universally accepted and serves as a foundation for understanding the properties of sets.

It is equally crucial to be aware that any other set with elements, for example, \( \{1, 2, 3\} \), can also be depicted using curly brackets with the elements separated by commas inside.

In the realm of set theory, the subset concept is a relationship between two sets. If all elements of set 'A' are also elements of set 'B', then we say that 'A' is a subset of 'B', symbolically represented as \( A \subseteq B \).

The subset relationship is a testament to the idea of inclusion within the set theory. For example, if we take the set of natural numbers \( \{1, 2, 3, 4, ...\} \) and a set of even numbers \( \{2, 4, 6, ...\} \), we can say that the set of even numbers is a subset of the set of natural numbers, as every even number is also a natural number.

Moreover, an empty set, denoted by \( \{\} \) or \( \varnothing \), is always a subset of any set, no matter what elements that set contains. This is because the definition of a subset does not require the subset to contain any elements but simply requires that every element of the empty set is also an element of the 'containing' set, which is trivially true.

Symbolic representation of sets allows us to communicate about sets in a precise and unambiguous way. With symbols, we can express complex relationships and operations involving sets. One of the fundamental symbols in set theory is the empty set symbol \( \varnothing \), which stands for a set with no elements.

Other common set symbols include 'U' for the union of two sets, '∩' for the intersection, and the backslash '\' for set difference. Using symbolic representation, mathematicians and students alike can easily understand the concepts of union, intersection, and difference. For example, if 'A' and 'B' are sets, then \( A \cup B \) (read as 'A union B') represents all elements that are in set 'A', set 'B', or in both.

Understanding and utilizing these symbols correctly is critical for studying set theory and comprehending the nature of mathematical sets. Each symbol is a compact way to express a potentially complicated idea, making it easier for students and mathematicians to work with sets in various mathematical contexts.