Final answer:
The true statements among the given are: a ∈ {a}, {3,5} = {5,3}, and {} = ∅. The false statements are: ∅ ∈ ∅, ∅ = {∅}, and ∅ ∈ {∅}.
Explanation:
The statement "a ∈ {a}" is true because it signifies that the element 'a' is an element of the set containing only 'a'. In set theory, a set can contain itself as an element, hence {a} includes 'a'.
The equality of sets {3,5} and {5,3} is true due to the commutative property of sets. The order of elements within a set doesn’t affect its identity, so both sets contain the same elements, albeit in different orders.
The statement "{} = ∅" is true as well. The empty set, denoted by ∅, is a set that contains no elements. Similarly, {} represents an empty set, and in set theory, both notations signify the same empty set.
Regarding the false statements, the statement "∅ ∈ ∅" is false. The empty set (∅) itself does not contain any elements, so it cannot be an element within itself according to the principles of set theory.
Similarly, the statement "∅ = {∅}" is false. ∅ represents an empty set, while {∅} denotes a set containing an empty set as its element. These are distinct notions in set theory.
Lastly, the statement "∅ ∈ {∅}" is false. This expression implies that the empty set (∅) is an element within a set that contains an empty set. However, in set theory, the empty set is distinct from a set containing the empty set as an element.