In abstract lattice theory, the symbol $leq$ is replaced with $subseteq$ in powerset lattice theory. Similarly, $vee$ and $wedge$ are replaced with $cup$ and $cap$, respectively, to translate from arbitrary lattices to lattices of sets. Regarding the question at hand, the author of an article stated that the lub of the empty set is $bot$, […]

# Category: Elementary set theory

## Proof of the Inclusion-Exclusion Principle for Three Sets

To begin with, the equation $|A cup B| = |A-B| + |B-A| + |A cap B|$ states that each segment on the right-hand side is separate. Using this information, we can rewrite the equation as follows: begin{eqnarray} |A cup B| + |A cap B| &=& |A-B| + |B-A| + |A cap B| + |A cap […]

## Why can you assume the existence of a bijection between naturals and rationals, but not between naturals and reals in Cantor’s Diagonalization Argument?

Kindly provide clarification. When considering a union, a countable set can be represented as a sequence. In the case of a cartesian product, each element can be viewed as a single element of the product of sequences. For example, the first tuple can be considered as the first element of S1, S2, S3, and so […]

## Distinctness of Set Elements Convention

I have experimented with different components for the drop-down feature, such as the DropDown, Gallery, and Menu. It is important to note that while these elements do not utilize keytips and resemble boxes, all items within a ribbon element must still be placed inside a group element initially. Question: In case we write something like: […]

## Which Symbol to Use for ‘If and Only If’ – $implies$ or $iff$?

To determine the truth or falsity of the statement $Aimplies B$, a truth table can be created. The table shows that if $A$ is true and $B$ is true, then the statement is true, but if $A$ is true and $B$ is false, then the statement is false. Likewise, if $Aimpliedby B$, it means that […]

## Understanding the Concept of Countably Infinite

Common Traces for Uncountable Sets are the following: the Cardinality is expressed as a power set of a set with infinite elements. Uncountable sets are those sets where it is impossible to list all the elements. In other words, there is no sequence that can list every element of the set at least once. Question: […]

## Symbolic representation of the highest value among a collection of functions

Solution 1: Note that $x^x = e^{xlog x}$ and minimizing $x^x$ is equivalent to minimizing $xlog x$. Solution 2: Let $f(x)=x^x$, which is only defined for $x>0$. Then $ln f(x)=xcdotln x$. By differentiating this expression and solving for $f^prime(x)$ using the chain rule and the product rule, we get $f^prime(x)=f(x)(1+ln x)=x^x(1+ln x)$. This can be […]

## Is the empty set {∅} a subset of every set including the set that contains it?

In Solution 2, it is mentioned that the natural numbers are built out of set theory by nesting empty sets inside other sets, which means that empty sets can be members of other sets. Moreover, considering that $mathcal{P}(A)subset Y$, the minimum cardinality that $Y$ could have is being questioned. To answer this, it is important […]

## Queries regarding Aleph-Aleph-Null

My understanding of the various values of ℵ was that they corresponded to the cardinalities of infinite sets. Specifically, ℵ₀ represented the cardinality of the set of all natural numbers. Additionally, if a set X had a cardinality of ℵₐ, then the cardinality of its powerset would be ℵₐ₊₁. However, the Wikipedia page on cardinal […]

## Confusion Surrounding Direct Proof: Is the Empty Set a Subset of All Sets?

In the construction of natural numbers using set theory, the number zero (represented by the empty set) is an element of every natural number greater than zero, and each of these natural numbers are sets. It may be possible to reconstruct the reasoning that leads to this false appearance, but the error is found in […]