What is the negation of " A ⊆ B "?

Hint: Use A ⊆ B ≡ ∀ x ( x ∈ A ⟹ x ∈ B )

Expert answers to "What is the negation of " A ⊆..."

mafalexpicsak

2022-10-15

What is the negation of "

A \subseteq B

Davin Meyer

Beginner2022-10-16Added 13 answers

Unfortunately not.

A \subset B

means

\forall x \in A, x \in B

, so the negation is

\exists x \in A, x \notin B

i.e. there is some element of

A

that is not in

B

.
Intuititively, since

A \subset B

means that

A

is "entirely contained in

B

". A not being a subset of

B

then means that

A

is not entirely contained in

B

. This includes the situation that they are disjoint, but also includes the situation where some elements of

A

are in

B

, but not all of them (i.e.

A

is partially contained in

B

Elise Kelley

Beginner2022-10-17Added 2 answers

Hint: Use

A \subseteq B \equiv \forall x (x \in A ⟹ x \in B)

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