Every singleton set has no proper subset. For example {6} has one and only one subset {6}which is improper subset.

Null set or empty set { } has no proper subset.

Write down the proper subset.

Given that | Two proper subset | possible subset |

{a,b,c} | {a }, {b} | 2³ = 8 |

{0,1} | {0},{1} | 2²=4 |

N = natural number | {1,2},{3,4,5} | infinite∴N={1,2,3…} |

R = real number | { },{4,5} | Infinite∴R = Q U Q’ |

{x/x ∈ Q∧ o<x ≤2} | {1}, {2} | Infinite |

(1) Difference Between {a,b} and {{a,b}} ?

Solution:

{a,b} is a set which has two element a and b, where as {{a,b}} is a singleton set whose element {a, b}.

(2) Set {a,b}={b,a} is true or false:

{a,b}={b,a}

is a true set

because arrangement of element in set does not matter

(3) Φ ⊆ {{a}}

True set

because empty set is a subset of every set.

singleton set has proper subset

(4) {a} ⊇ {{a}}

False

As {{a}} is a singleton set having element {a}

But {a} is not a super set of {{a}}

(5) {a} ∈ {{a}}

TRUE

As {{a}} is a singleton set having element {a}

(6) a ∈ {{a}}

FALSE

As {{a}} is a singleton set having element {a}. But a ∈ {a}.

(7) φ ∈ {{a}}

FALSE

As {{a}} is a singleton set having element {a}. But φ ⊆ {{a}}