Question

(i) Determine whether each of these statements is true or false.

(a) 0∈∅
(b)∅∈{0}
(c){0}⊂∅
(d)∅⊂{0}
(e){0}∈{0}
(f){0}⊂{0}
(g){∅}⊆{∅}
(ii) Determine whether these statements are true or false.
(a)∅∈{∅}
(b)∅∈{∅,{∅}}
(c){∅}∈{∅}
(d){∅}∈{{∅}}
(e){∅}⊂{∅,{∅}}
(f){{∅}}⊂{∅,{∅}}
(g){{∅}}⊂{{∅},{∅

Answer 1

In this question, we are given various statements and we need to determine whether they are true or false. We also need to correct any false statements.

Step-by-step explanation:

(i) Determine whether each of these statements is true or false.
(a) 0∈∅ - False, because 0 is not an element of the empty set.
(b) ∅∈{0} - False, because the empty set is not an element of the set containing 0.
(c) {0}⊂∅ - False, because the set containing 0 is not a subset of the empty set.
(d) ∅⊂{0} - True, because the empty set is a subset of any set.
(e) {0}∈{0} - True, because the set containing 0 is an element of itself.
(f) {0}⊂{0} - True, because the set containing 0 is a subset of itself.
(g) {∅}⊆{∅} - True, because the set containing the empty set is a subset of the set containing the empty set.

(ii) Determine whether these statements are true or false.
(a) ∅∈{∅} - True, because the empty set is an element of the set containing the empty set.
(b) ∅∈{∅,{∅}} - True, because the empty set is an element of the set containing the empty set and another set.
(c) {∅}∈{∅} - False, because the set containing the empty set is not an element of the set containing the empty set.
(d) {∅}∈{{∅}} - True, because the set containing the empty set is an element of the set containing another set.
(e) {∅}⊂{∅,{∅}} - True, because the set containing the empty set is a subset of the set containing the empty set and another set.
(f) {{∅}}⊂{∅,{∅}} - True, because the set containing a set containing the empty set is a subset of the set containing the empty set and another set.
(g) {{∅}}⊂{{∅},{∅}} - False, because the set containing a set containing the empty set is not a subset of the set containing two empty sets.