. 0 ∈ ∅ 2. ∅ ∈ {0} 3. {0} ⊆ ∅4. ∅ ⊂ {0} 5. {∅} ⊆ {∅} 6. {0} ⊆ {0}7. ∅ ∈ {∅} 8. {∅} ∈ {{∅}} 9. {∅} ⊂ {∅, {∅}}10. x ∈ {x} 11. ∅ ⊆ {x} 12. ∅ ∈ {x}Lec 2 TS 5
Question
1. 0 ∈ ∅
2. ∅ ∈ {0}
3. {0} ⊆ ∅
4. ∅ ⊂ {0}
5. {∅} ⊆ {∅}
6. {0} ⊆ {0}
7. ∅ ∈ {∅}
8. {∅} ∈ {{∅}}
9. {∅} ⊂ {∅, {∅}}
10. x ∈ {x}
11. ∅ ⊆ {x}
12. ∅ ∈ {x}
Lec 2 TS 5
Solution
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0 ∈ ∅: False. The empty set ∅ does not contain any elements, including 0.
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∅ ∈ {0}: False. The set {0} contains the element 0, not the empty set ∅.
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{0} ⊆ ∅: False. The empty set ∅ does not contain any elements, so it cannot contain the set {0}.
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∅ ⊂ {0}: True. The empty set ∅ is a subset of any set, including {0}.
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{∅} ⊆ {∅}: True. The set {∅} is a subset of itself.
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{0} ⊆ {0}: True. Any set is a subset of itself.
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∅ ∈ {∅}: True. The set {∅} contains the empty set ∅ as an element.
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{∅} ∈ {{∅}}: True. The set {{∅}} contains the set {∅} as an element.
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{∅} ⊂ {∅, {∅}}: True. The set {∅} is a proper subset of the set {∅, {∅}}.
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x ∈ {x}: True. Any element x is an element of the set {x}.
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∅ ⊆ {x}: True. The empty set ∅ is a subset of any set, including {x}.
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∅ ∈ {x}: False. The set {x} contains the element x, not the empty set ∅.
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