Verification of my set solutions Can you verify my proof if it is right?  Let A and B be sets. (a) Show that A is a subset of B if and only if for any set C, one has A U C is a subset of B U C. (b) Show that A is a subset of B if and only if for any set C, one has A intersect C is a subset of B intersect C.  (a)   To prove (=>) Given A is a subset of B, this implies A U B = B. Given a set C, A U B U C = B U C. Since A is a subset of B, this implies A U C is a subset of B U C. To prove (&lt;=) Solution 1 Given A U C is a subset of B U C , For A U C , x is a member of A U C . For B U C, x is a member of B U C. Suppose x is a member of A but not a member of B, meaning A is not a subset of B. Therefore , A U C is not a subset of B U C. Hence, A must be a subset of B. To prove (&lt;=) Solution 2 Given A U C is a subset of B U C, For A U C, x is a member of A U C and for B U C, x is a member of B U C. Suppose C is a disjoint set. Since A U C is a subset of B U C, it follows that A is a subset of B.   (b)    To prove (=>) Since A is a subset of B , it follows that A U B = B. A intersect C is a subset of A by definition of intersection. B intersect C is a subset of B by definition of intersection. Let x be member of A and member of C. Since A intersect C is a subset of A, x must be in A. Let x be a member of B and member of C. Since B intersect C is a subset of B, x must be in B. Therefore, A intersect C is a subset of B intersect C. To prove (&lt;=) Since A intersect C is a subset of B intersect C, suppose we let Set C be a subset of B and a subset of A as well.  C is a subset of A implies A intersect C is equals to C. C is a subset of B implies B intersect C is equals to C as well. Since A intersect C is equals to B intersect C, A is a subset of B.
Let A and B be sets. Show that A is a subset of B if and only if for any set C, one has A union C is a subset of B union C. Can you verify my proof if it is right?  Let $A$ and $B$ be sets.  (a) Show that $A$ is a subset of $B$ if and only if for any set $C$, one has $A$ union $C$ is a subset of $B$ union $C$. (b) Show that $A$ is a subset of $B$ if and only if for any set $C$, one has $A$ intersect $C$ is a subset of $B$ intersect $C$.  Edited Solution :  (a)  (=>) Given A is a subset of B, this implies A U B = B. Given a set C, A U B U C = B U C. Since A is a subset of B, this implies A U C is a subset of B U C.      Solution 1 :(&lt;=) Given A U C is a subset of B U C , For A U C , x is a member of A U C . For B U C, x is a member of B U C. Suppose x is a member of A but not a member of B, meaning A is not a subset of B. Therefore , A U C is not a subset of B U C. Hence, A must be a subset of B.      Solution 2 :(&lt;=) Given A U C is a subset of B U C, For A U C, x is a member of A U C and for B U C, x is a member of B U C. Suppose C is a disjoint set. Since A U C is a subset of B U C, it follows that A is a subset of B.  (b) (=>) Since A is a subset of B , it follows that A U B = B. [A intersect C] is a subset of A and  [B intersect C ] is a subset of B by definition of intersection. Let x be member of A and member of C. Since [ A intersect C ] is a subset of A, x must be in A. Let x be a member of B and member of C. Since [ B intersect C ] is a subset of B, x must be in B. Therefore, A intersect C is a subset of B intersect C.  (&lt;=) Since A intersect C is a subset of B intersect C, suppose we let Set C be a subset of B and a subset of A as well. C is a subset of A implies A intersect C is equals to C. C is a subset of B implies B intersect C is equals to C as well. Since A intersect C is equals to B intersect C, A is a subset of B.
Database program with CSV import and ad-hoc SQL queries? Is there a database program for the Mac that makes it easy to import a CSV file into a table and then query and manipulate the data using ad-hoc SQL queries?  (A Windows example of a similar product would be Microsoft Access ... it also has forms and reports, which are not as important, but I would not mind a Mac solution that has those too.) My screen has 1280 x 1024 resolution. But unfortunately, rightmost belt of 200 pixels is damaged and unable do display any image.  How can I set that the Xorg will treat my screen as 1080 x 1024 device?  Now I have tried:  Virtual 1080 1024 Viewport 0 0   It doesn't work right - image is scaled to fit vertically, and horizontally I can pan with the mouse. How does ServerName and ServerAlias work? Imagine a situation in which the whole place is on fire, a bomb is about to explode, everyone is running for their lives and someone is checking his looks on the mirror... pretty inappropriate for the situation, don't you think? I'm looking for an idiom or colloquial/slang/informal expression to describe such an indifferent stance. When getting a new credit card, what is the policy in regards to signing the strip on the back? Should I sign it? Should I write "See ID"? Should I leave it blank? Should I sign it AND write "See ID"?   I see all sorts of different advice but it seems like it's all based on urban legends, hearsay, or self interest (merchants and banks seem to advise you to do what's best for them, not what's best for the customer).  My question: For an average everyday consumer who uses a card fairly frequently, what is the best practice? What practice is most secure? The program ed, a minimal text editor, cannot be exited by sending it an interrupt through using Ctrl-C, instead printing the error message "?" to the console.  Why doesn't ed just exit when it receives the interrupt?  Surely there's no reason why a cryptic error message is more useful here than just exiting.  This behavior leads many new users into the following sort of interaction:   $ ed hello ? help ? exit ? quit ? ^C ? ^C ? ? ? ^D $ su # rm -f /bin/ed    Such a tragic waste—easily avoidable if ed simply agreed to be interrupted.  Another stubborn program exhibiting similar behavior is less which also doesn't appear to have much reason to ignore C-c.  Why don't these programs just take a hint? Regular expression: match start or whitespace Can a regular expression match whitespace or the start of a string?  I'm trying to replace currency the abbreviation GBP with a £ symbol. I could just match anything starting GBP, but I'd like to be a bit more conservative, and look for certain delimiters around it.  &gt;&gt;&gt; import re &gt;&gt;&gt; text = u'GBP 5 Off when you spend GBP75.00'  &gt;&gt;&gt; re.sub(ur'GBP([\W\d])', ur'£\g&lt;1&gt;', text) # matches GBP with any prefix u'\xa3 5 Off when you spend \xa375.00'  &gt;&gt;&gt; re.sub(ur'^GBP([\W\d])', ur'£\g&lt;1&gt;', text) # matches at start only u'\xa3 5 Off when you spend GBP75.00'  &gt;&gt;&gt; re.sub(ur'(\W)GBP([\W\d])', ur'\g&lt;1&gt;£\g&lt;2&gt;', text) # matches whitespace prefix only u'GBP 5 Off when you spend \xa375.00'   Can I do both of the latter examples at the same time? How can I find the possible values that $\gcd(a+b,a^2+b^2)$ can take, if $\gcd(a,b)=1$ Where can I get the original Unix? Objects with transparent texture not always show transparency in 3dview For some reason objects don't show their transparency all the time.  In this image the circles in red are suposed to be transparent like the ones in green.    When I select the object in the back, all objects in front of it become opaque.    When I select the two objects that are opaque, then they become transparent as they sould be.    The display is in GLSL mode  These are the material and texture properties      I'm not worried about the final render, just the editing view. I would like to see the actual transparency at editing time only, because the scene is to be imported in a third party game engine.  Blender is v2.68   is the blend file Zero-pad digits in string Why isn't temperature frame dependent? In (non-relativistic) classical physics, if the temperature of an object is proportional to the average kinetic energy ${1 \over 2} m\overline {v^{2}}$of its particles (or molecules), then shouldn't that temperature depend on the frame of reference - since $\overline {v^{2}}$ will be different in different frames?  (I.e. In the lab frame  $K_l = {1 \over 2} m\overline {v^{2}} $, but in a frame moving with velocity $u$ relative to the lab frame, $K_u =  {1 \over 2} m \overline {(v+u)^{2}}$). Git keeps prompting me for a password Why are there exactly four nucleobases in DNA? Does someone know why DNA is composed of four nucleobases? In particular, is there an explanation for the number? Why four and not two, or eight? Is the set of all finite sequences of letters of Latin alphabet countable/uncountable? How to prove either? Post a Form to Popup window not to the parent How do I change my update server in Ubuntu 18.04? The first thing I did after installing 16.04 LTS was to change my update server to Main Server. I can not find a similar option in Ubuntu 18.04.  How do I change it? Compact operators on an infinite dimensional Banach space cannot be surjective I am reading a book about functional analysis and have a question:      Let $X$ be a infinite-dimensional Banach-space and $A:X \rightarrow X$ a compact operator. How can one show that $A$ can not be surjective? If two functionals have the same kernel, then one is a multiple of the other I would like some help with this exercise.  Suppose that $f_1,\ f_2 \in V^*$ and that $\text{Ker} f_1 = \text{Ker} f_2$. Show that $f_1 = k f_2$ for some scalar $k$.  I expect your suggestions.  Thanks! I'm working on a 2nd order passive low pass filter, consisting of two passive low pass filters chained together.         &ndash; Schematic created using   Let \$ H(s) = H_1(s)H_2(s) \$ where \$ H_1(s) \$ and \$ H_2(s) \$ are the transfer functions for each separate filter stage.  Then \$ |H(s)| = |H_1(s)||H_2(s)| \$  Knowing the magnitude of a passive low pass filter,   $$|H(s)| = \dfrac1{\sqrt{ (\omega R_1C_1)^2 + 1} } \times \dfrac1{\sqrt{ (\omega R_2C_2)^2 + 1} } = \dfrac1{\sqrt{((\omega R_1C_1)^2 + 1)((\omega R_2C_2)^2 + 1)}} $$  Then trying to find the cutoff frequency:  $$ \left(\dfrac1{\sqrt{2}}\right)^2  = \dfrac1{\sqrt{((\omega R_1C_1)^2 + 1)((\omega R_2C_2)^2 + 1)}} $$ $$ 2  = \sqrt{((\omega R_1C_1)^2 + 1)((\omega R_2C_2)^2 + 1)} $$ $$ 4  = ((\omega R_1C_1)^2 + 1)((\omega R_2C_2)^2 + 1) $$ $$ 4  = (\omega R_1C_1)^2(\omega R_2C_2)^2 + (\omega R_1C_1)^2 + (\omega R_2C_2)^2 + 1 $$  And I'm stuck. Research on the web tells me \$ \omega_c = \dfrac1{\sqrt{R_1C_1R_2C_2}} \$, but I can't find why? Can anybody show my the derivation to find this?