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

1)

a)0 $\in\varnothing$

Ans : False

Empty set does not contain any elements

b)$\varnothing$ $\isin$ {0}

Empty set is an element of set that contains only zero

So answer is false

c)

{0}$\subset$ $\varnothing$

The empty set that contains nothing so set zero is not the subset of empty set

d)$\varnothing\subset$ {0}

The set zero contains empty set so empty set is subset of zero set

Answer is true

e){0}$\isin$ {0}

The set containing zero element so zero elements belongs to set of zero not zero set{0}

So answer is false

F){0}$\subset$ {0}

A set is always not a subset of itself ,because the sets are completely identical

Given statment is false

G){0} subset {0}

Set is subset of itself

Answer is true

2)

a)$\varnothing\isin$ {$\varnothing$ }

The empty set contains empty, so empty strings belongs to empty set

So true

b)

$\varnothing\isin$ {$\varnothing$ ,{$\varnothing$ }}

The empty elements belongs to set and subset of empty

So answer is true

c){$\varnothing$ }$\isin$ {$\varnothing$ }

The set containing elements, not containing sets

So answer is false

d){$\varnothing$ }$\isin$ {{$\varnothing$ }}

The empty set is an element of set

So answer is true

e){ $\varnothing$}$\subset$ {$\varnothing$ ,{$\varnothing$ }}

The empty string is subset of given set

So true

f){{$\varnothing$ }}$\subset$ {$\varnothing$ ,{$\varnothing$ }}

The empty string is subset of given set

So answer is true

G){{$\varnothing$ }}$\subset$ {{$\varnothing$} ,{$\varnothing$}}

Answer is false

Proper set not equal sets