Final answer:
The statements (a) 'x ∈ {x}' and (b) '{x} ⊆ {x}' are true, (c) '{x} ∈ {x}' is false and should be '{x} ∈ {{x}}', and (d) '∅ ⊆ {x}' is true.
Step-by-step explanation:
We will evaluate the truth value of each of the following statements regarding set theory and elements of a set:
- a. x ∈ {x} - This statement is true. It means that the element x is a member of the set containing only x.
- b. {x} ⊆ {x} - This statement is true. It says that the set containing x is a subset of itself, which is always true for any set.
- c. {x} ∈ {x} - This statement is false. The correct statement is {x} ∈ {{x}}, meaning a set containing x is an element of a set containing the set containing x.
- d. ∅ ⊆ {x} - This statement is true. It declares that the empty set is a subset of the set containing x, which is a fundamental principle in set theory.