Determine whether the following statements are true or...

Determine whether the following statements are true or false. (a) If $\{1\} \in \mathcal{P}(A)$, then $1 \in A$ but $\{1\} \notin A$. (b) If $A, B$ and $C$ are sets such that $A \subset \mathcal{P}(B) \subset C$ and $|A|=2$, then $|C|$ can be 5 but $|C|$ cannot be 4 . (c) If a set $B$ has one more element than a set $A$, then $\mathcal{P}(B)$ has at least two more elements than $\mathcal{P}(A)$. (d) If four sets $A, B, C$ and $D$ are subsets of $\{1,2,3\}$ such that $|A|=|B|=|C|=|D|=2$, then at least two of these sets are equal.

Determine whether the following statements are true or false.
(a) If $\{1\} \in \mathcal{P}(A)$, then $1 \in A$ but $\{1\} \notin A$.
(b) If $A, B$ and $C$ are sets such that $A \subset \mathcal{P}(B) \subset C$ and $|A|=2$, then $|C|$ can be 5 but $|C|$ cannot be 4 .
(c) If a set $B$ has one more element than a set $A$, then $\mathcal{P}(B)$ has at least two more elements than $\mathcal{P}(A)$.
(d) If four sets $A, B, C$ and $D$ are subsets of $\{1,2,3\}$ such that $|A|=|B|=|C|=|D|=2$, then at least two of these sets are equal.

Key Concepts

Pigeonhole Principle

The pigeonhole principle is a combinatorial tool used to deduce the inevitability of repetition or overlapping when a finite number of objects are distributed among fewer containers. In set theory, it is applied to argue about the inevitable existence of identical subsets when considering constraints like fixed numbers of elements within subsets.

Subset Relations and Inclusion

Understanding subset relations, including proper and improper subsets, is key in problems that involve chains of inclusions (like A ? P(B) ? C). These relations impose restrictions on the elements and possible cardinalities of the sets involved.

Cardinality and Exponential Growth

Cardinality is the measure of the number of elements in a set. This concept is crucial when comparing sizes of sets and their power sets. The transformation from a set to its power set is characterized by exponential growth, as every element of the set contributes to doubling the number of subsets.

Element vs Subset

It is important to distinguish between an element and a subset. An element is an individual member of a set, while a subset is a set composed of elements from the original set. For example, even if a singleton set {x} might appear related to x, {x} as a subset of a set is not the same as x being an element of that set.

Power Set

The power set of a set, denoted by P(A), is the set of all subsets of A. This concept is fundamental because it introduces exponential growth in cardinality; for a set A with n elements, the power set has 2^n elements. Understanding power sets also helps differentiate between an element of a set and a subset of a set.

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