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Determine which of the following statements are true, and which are false.

a∈{a}
{3,5}={5,3}
∅∈∅
∅={∅}
{}=∅
∅∈{∅}

User Brad Parks
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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.

User Tej Chajed
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