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

Question

-Dominant-

14 days ago

14

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

) {{∅}} ⊂ {∅, {∅}} g) {{∅}} ⊂ {{∅}, {∅}}

Mathematics

1 answer:

zzz [4K] · Answer 1 · 2026-02-19T00:23:52+03:00

Answer:

a) True

b) True

c) False

d) True

e) True

f) True

g) True

Step-by-step breakdown:

Let A denote a set. The element a is part of A if and only if a∈A.

a) The sole element of {∅} is ∅, so it follows that ∅∈{∅}.

b) The elements of {∅, {∅}} are ∅ and {∅}, thus ∅ ∈ {∅, {∅}}.

c) As {∅} solely contains ∅, it is clear {∅}≠∅ due to {∅} having one element while ∅ has no elements. Hence, {∅} ∉ {∅} since {∅} is not an element of {∅}.

d) The one element in {{∅}} is {∅}. Thus, {∅} ∈ {{∅}}.

e) The items in {∅, {∅}} include ∅ and {∅}. The only element of {∅} is ∅. Thus, every element of {∅} appears in {∅, {∅}} too, leading to {∅} ⊂ {∅, {∅}}.

f) With elements of {∅, {∅}} being ∅ and {∅}, and since the only element of {{∅}} is {∅}, we determine that {{∅}} appears in {∅, {∅}}, which means {{∅}} ⊂ {∅, {∅}}.

g) The only element of {{∅}, {∅}} equals {{∅}} which is {∅}. Each element of {{∅}} is also within {{∅}}, therefore {{∅}} ⊂ {{∅}, {∅}}={{∅}}.