Solve both parts and justify the conditions used. Part A: Prove that the empty set is a subset of every set. Part B: How many subsets does A have?

01 / Standard#Your turn

\begin{gathered}\text{A: }\varnothing\subseteq A\\\text{B: }|A|=10\end{gathered}

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

A: Apply the element-by-element definition of inclusion.

Hint 2

B: Apply the element-by-element definition of inclusion.

Worked solution

Part A reasoning
Apply the element-by-element definition of inclusion.
Calculate or simplify this relation.
$\neg(\varnothing\subseteq A)\Rightarrow\exists x\in\varnothing:x\notin A$
The proposed counterexample cannot exist.
Part B reasoning
Apply the element-by-element definition of inclusion.
Calculate or simplify this relation.
$2^{10}=1024$
The empty set and the full set both count.

A: No element of the empty set can violate inclusion. B: The requested value is 1024.

Checks and common pitfalls: The proposed counterexample cannot exist. The empty set and the full set both count.

Reasoning checklist · self / teacher assessment

Think first. Reveal a hint when the class is ready.

Sets and commonly used logic: mixed assessment