Determine the truth values of the following propositions. Explain.a) ∅ ∈ {1}b) {1} ∈ {1}c) {1} ⊂ {1}d) {1} ∈ {1, {1}}e) {1} ⊆ {1, {1}}f) {{1}} ∈ 𝒫({1, {1}})
Question
Determine the truth values of the following propositions. Explain.
a) ∅ ∈ {1}
b) {1} ∈ {1}
c) {1} ⊂ {1}
d) {1} ∈ {1, {1}}
e) {1} ⊆ {1, {1}}
f) {{1}} ∈ 𝒫({1, {1}})
Solution
a) ∅ ∈ {1} - False. The empty set is not an element of the set containing 1.
b) {1} ∈ {1} - False. The set containing 1 is not an element of the set containing 1. It is a subset.
c) {1} ⊂ {1} - False. The set containing 1 is not a proper subset of the set containing 1. It is a subset, but not a proper subset because there are no elements in {1} that are not in {1}.
d) {1} ∈ {1, {1}} - True. The set containing 1 is an element of the set containing 1 and the set containing 1.
e) {1} ⊆ {1, {1}} - True. The set containing 1 is a subset of the set containing 1 and the set containing 1.
f) {{1}} ∈ 𝒫({1, {1}}) - True. The set containing the set containing 1 is an element of the power set of the set containing 1 and the set containing 1. The power set includes all subsets, including the empty set, the set itself, and all combinations of elements, which in this case includes the set containing the set containing 1.
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