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The (there exists) symbol is used in math to express the existence of a variable. For example, the symbol is usually used in an expression like this.
In plain language, this expression means there exists a variable belonging to the set of natural numbers such that is even.
The member of symbol indicates that an expression belongs to or is in the set.
The colon symbol is used in math to represent a "such that" statement and also to represent a ratio in mathematics.
The "there does not exist" symbol.
The syntax for "there exists" is a backwards captial E. It is often used in conjuction with a variable with certain properties.