De Morgan's Laws (Set Theory)/Set Difference/General Case/Difference with Intersection

From ProofWiki
Jump to navigation Jump to search

Theorem

Let $S$ and $T$ be sets.

Let $\powerset T$ be the power set of $T$.

Let $\mathbb T \subseteq \powerset T$.


Then:

$\displaystyle S \setminus \bigcap \mathbb T = \bigcup_{T' \mathop \in \mathbb T} \paren {S \setminus T'}$

where:

$\displaystyle \bigcap \mathbb T := \set {x: \forall T' \in \mathbb T: x \in T'}$

that is, the intersection of $\mathbb T$


Proof

Suppose:

$\displaystyle x \in S \setminus \bigcap \mathbb T$

Note that by Set Difference is Subset we have that $x \in S$ (we need this later).

Then:

\(\displaystyle x\) \(\in\) \(\displaystyle S \setminus \bigcap \mathbb T\)
\(\displaystyle \leadstoandfrom \ \ \) \(\displaystyle x\) \(\notin\) \(\displaystyle \bigcap \mathbb T\) Definition of Set Difference
\(\displaystyle \leadstoandfrom \ \ \) \(\displaystyle \neg (\forall T' \in \mathbb T: (x\) \(\in\) \(\displaystyle T'))\) Definition of Intersection of Set of Sets
\(\displaystyle \leadstoandfrom \ \ \) \(\displaystyle \exists T' \in \mathbb T: \neg (x\) \(\in\) \(\displaystyle T')\) De Morgan's Laws (Predicate Logic)
\(\displaystyle \leadstoandfrom \ \ \) \(\displaystyle \exists T' \in \mathbb T: x\) \(\in\) \(\displaystyle S \setminus T'\) Definition of Set Difference: note $x \in S$ from above
\(\displaystyle \leadstoandfrom \ \ \) \(\displaystyle x\) \(\in\) \(\displaystyle \bigcup_{T' \mathop \in \mathbb T} \paren {S \setminus T'}\) Definition of Union of Set of Sets


Therefore:

$S \setminus \bigcap \mathbb T = \bigcup_{T' \mathop \in \mathbb T} \paren {S \setminus T'}$

$\blacksquare$


Caution

It is tempting to set up an argument to prove the general case using induction. While this works, and is a perfectly valid demonstration for an elementary student in how such proofs are crafted, such a proof is inadequate as it is valid only when $\mathbb T$ is finite.

The proof as given above relies only upon De Morgan's laws as applied to predicate logic. Thus the uncountable case has been reduced to a result in logic.

However, for better or worse, the following is an example of how one might achieve this result using induction.


Proof by Induction

Let the cardinality $\card I$ of the indexing set $I$ be $n$.

Then by the definition of cardinality, it follows that $I \cong \N^*_n$ and we can express the proposition:

$\displaystyle S \setminus \bigcap_{i \mathop \in I} T_i = \bigcup_{i \mathop \in I} \paren {S \setminus T_i}$

as:

$\displaystyle S \setminus \bigcap_{i \mathop = 1}^n T_i = \bigcup_{i \mathop = 1}^n \paren {S \setminus T_i}$

The proof of this is more amenable to proof by Principle of Mathematical Induction.


For all $n \in \N_{>0}$, let $\map P n$ be the proposition:

$\displaystyle S \setminus \bigcap_{i \mathop = 1}^n T_i = \bigcup_{i \mathop = 1}^n \paren {S \setminus T_i}$


$\map P 1$ is true, as this just says $S \setminus T_1 = S \setminus T_1$.


Base Case

$\map P 2$ is the case:

$S \setminus \paren {T_1 \cap T_2} = \paren {S \setminus T_1} \cup \paren {S \setminus T_2}$

which has been proved.

This is our basis for the induction.


Induction Hypothesis

Now we need to show that, if $\map P k$ is true, where $k \ge 2$, then it logically follows that $\map P {k + 1}$ is true.


So this is our induction hypothesis:

$\displaystyle S \setminus \bigcap_{i \mathop = 1}^k T_i = \bigcup_{i \mathop = 1}^k \paren {S \setminus T_i}$


Induction Step

Now we need to show:

$\displaystyle S \setminus \bigcap_{i \mathop = 1}^{k + 1} T_i = \bigcup_{i \mathop = 1}^{k + 1} \paren {S \setminus T_i}$


This is our induction step:

\(\displaystyle S \setminus \bigcap_{i \mathop = 1}^{k + 1} T_i\) \(=\) \(\displaystyle S \setminus \paren {\bigcap_{i \mathop = 1}^k T_i \cap T_{k + 1} }\) Intersection is Associative
\(\displaystyle \) \(=\) \(\displaystyle \paren {S \setminus \bigcap_{i \mathop = 1}^k T_i} \cup \paren {S \setminus T_{k + 1} }\) Base case
\(\displaystyle \) \(=\) \(\displaystyle \paren {\bigcup_{i \mathop = 1}^k \paren {S \setminus T_i} } \cup \paren {S \setminus T_{k + 1} }\) Induction hypothesis
\(\displaystyle \) \(=\) \(\displaystyle \bigcup_{i \mathop = 1}^{k + 1} \paren {S \setminus T_i}\) Union is Associative


So $\map P k \implies \map P {k + 1}$ and the result follows by the Principle of Mathematical Induction.


Therefore:

$\displaystyle S \setminus \bigcap_{i \mathop = 1}^n T_i = \bigcup_{i \mathop = 1}^n \paren {S \setminus T_i}$

that is:

$\displaystyle S \setminus \bigcap_{i \mathop \in I} T_i = \bigcup_{i \mathop \in I} \paren {S \setminus T_i}$

$\blacksquare$


Source of Name

This entry was named for Augustus De Morgan.


Sources