$$    [\exists x \in U, P(x) \land Q(x)]   \Leftrightarrow [(\exists x \in U, P(x)) \land (\exists x \in U, Q(x))] $$
Context  I'm having trouble understanding the limited situations in which qualifiers can be distributed.  I am given that the rules are:  $$\forall x\left[P(x)\land Q(x)\right]\equiv\forall xP(x) \land\forall xQ(x) \tag{1}$$ $$\exists x\left[P(x)\lor Q(x)\right]\equiv\exists xP(x) \lor\exists xQ(x) \tag{2}$$  $$\forall x\left[P(x)\lor Q(x)\right]\not\equiv\forall xP(x) \lor\forall xQ(x) \tag{3}$$ $$\exists x\left[P(x)\land Q(x)\right]\not\equiv\exists xP(x) \land\exists xQ(x) \tag{4}$$    Questions   What are examples of $P(x)$ and $Q(x)$ that illustrate when these statements are true? How can I show these statements always hold using proofs? Why can't "$\forall x$" be distributed in example 3 or "$\exists x$" in example 4?
Why use a prime number in hashCode? In Chiswell's Mathematical Logic, there is the following problem is posed using the language of propositions (LP):     In some systems of logic (mostly constructive systems where 'true' is taken to mean 'provable') there is a rule      if $(\Gamma\vdash(\phi\,\lor\psi))$ is a correct sequent then at least   one of $(\Gamma\vdash\phi)$ and $(\Gamma\vdash\psi)$ is also correct.      By giving a counterexample to a particular instance, show that this is   unacceptable as a rule for LP. [Start by giving counterexamples for   both the sequents $(\vdash p_0)$ and $(\vdash(\neg p_0))$.]   In my , it was answered that one can let $\Gamma = \{p_0\,\lor\,(\neg p_0)\}$.  From there, one can deduce neither $(\Gamma\vdash p_0)$ nor $(\Gamma\vdash (\neg p_0))$.  But this does not proceed by the counterexample route that the author suggests.  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I just want a very simple prime generator (nothing fancy).   import math  def main():     count = 3     one = 1     while one == 1:         for x in range(2, int(math.sqrt(count) + 1)):             if count % x == 0:                  continue             if count % x != 0:                 print count          count += 1 Is a Mexican passport needed to fly into Mexico from the United States? Enumerate non-isomorphic graphs on n vertices In the following the graphs are assumed to be undirected and simple.  1.Enumerate the number of non-isomorphic graphs on $n$ vetrices where $n$ is fixed.  Here are some ideas I had:  The number of labeled graphs is $ 2^{\frac{n(n-1)}{2}} $.  So it is enough to find the number unlabeled graphs on $n$ vertices.I have no idea for this.  2.Enumerate the number of non-isomorphic graphs on $n$ vertices and $m$ edges where $n,m$ are fixed.   Can we find a closed formula for each of this?  Any help?   Thank you! 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Or any extrapolation of how long Luke's journey on Dagobah was? Was it a day? Several weeks? Some months? Search for a string in all tables, rows and columns of a DB Beamer continue block in next column For any smooth manifold, is it true that for any two points on the manifold, there exists a chart that covers the two points? Some say that we can connect two points with a continuous curve and a small contractible neighborhood of the path together with the charts of the points can be regarded as the chart. But I don't know how to verify this statement. TTY doesn't wait for password Get number of unread sms How do I send a file as an email attachment using Linux command line? I am trying to capitalize Western Canada or western Canada properly and am wanting a definition for when the "ern" is added as a suffix to a locational distinction of a proper noun.  I believe the capitalization is related to the suffix and am grouping two questions.  After researching on the internet I have found a trend that areas of discrete area become capitalized along with their proper noun   East New York West Hollywood East London (Britain) East Los Angeles   Some locations have "ern" added to their locational modifier.    Web Examples (from government sites):   eastern Canada ( Western Canada ()   All the examples listed contain discrete areas.  The "ern" suffix is used for larger areas, but I am unable to determine when an area becomes large enough. Normal subgroups of infinite symmetric group I recently took a course on group theory, which mentioned that the following proposition is equivalent to the continuum hypothesis: "The infinite symmetric group (i.e. the group of permutations on the set $\mathbb{N}$) has exactly 4 normal subgroups." Does anyone have any references or explanation for this?