Question

A set X is said to be a proper subset of Y if and
only if​

Answer 1

Answer:

Step-by-step explanation:

"Superset" redirects here. For other uses, see Superset (disambiguation).

"⊃" redirects here. For the logic symbol, see horseshoe (symbol). For other uses, see horseshoe (disambiguation).

Euler diagram showing

A is a proper subset of B,  A⊂B,  and conversely B is a proper superset of A.

In mathematics, a set A is a subset of a set B if all elements of A are also elements of B; B is then a superset of A. It is possible for A and B to be equal; if they are unequal, then A is a proper subset of B. The relationship of one set being a subset of another is called inclusion (or sometimes containment). A is a subset of B may also be expressed as B includes (or contains) A or A is included (or contained) in B.

The subset relation defines a partial order on sets. In fact, the subsets of a given set form a Boolean algebra under the subset relation, in which the join and meet are given by intersection and union, and the subset relation itself is the Boolean inclusion relation.

If A and B are sets and every element of A is also an element of B, then:

A is a subset of B, denoted by {\displaystyle A\subseteq B,}{\displaystyle A\subseteq B,} or equivalently

B is a superset of A, denoted by {\displaystyle B\supseteq A.}B\supseteq A.[1]

If A is a subset of B, but A is not equal to B (i.e. there exists at least one element of B which is not an element of A), then:

A is a proper (or strict) subset of B, denoted by {\displaystyle A\subsetneq B}A\subsetneq B (or {\displaystyle A\subset B}A\subset B[1][2]). Or equivalently,

B is a proper (or strict) superset of A, denoted by {\displaystyle B\supsetneq A}{\displaystyle B\supsetneq A} (or {\displaystyle B\supset A}B \supset A[1]).

The empty set, written { } or ∅, is a subset of any set X and a proper subset of any set except itself.

For any set S, the inclusion relation ⊆ is a partial order on the set {\displaystyle {\mathcal {P}}(S)}{\mathcal {P}}(S) (the power set of S—the set of all subsets of S[3]) defined by {\displaystyle A\leq B\iff A\subseteq B}{\displaystyle A\leq B\iff A\subseteq B}. We may also partially order {\displaystyle {\mathcal {P}}(S)}{\mathcal {P}}(S) by reverse set inclusion by defining {\displaystyle A\leq B\iff B\subseteq A.}{\displaystyle A\leq B\iff B\subseteq A.}

When quantified, A ⊆ B is represented as ∀x(x ∈ A → x ∈ B).[4]