Elements of set theory
This part will cover
- Unique
- Union
- Set difference
- Intersection
You are a system administrator aboard the corporate OC-3. You are currently deploying a database change affecting 4423 customers at a speed of 155.52Mbit/s.
In brief intervals of time spent waiting between helping your sales staff with Excel, you enjoy maintaining a forum dedicated to signal processing. Unfortunately for you, you had a lapse of judgement and your forum's been completely spammed. Unfortunately for management, you’ve decided to use company time to fix this issue.
POSTS
┌──────────┬─────┬────────────────────────────────────────────────────────────────────┬────────────────┐
│30-08-2024│18:52│Why does DVB-C use QAM instead of OFDM? │frequencySniffer│
├──────────┼─────┼────────────────────────────────────────────────────────────────────┼────────────────┤
│30-08-2024│10:34│DAC in QPSK modulation │radioComputer │
├──────────┼─────┼────────────────────────────────────────────────────────────────────┼────────────────┤
│30-08-2024│02:30│We’ve Received Your Payment │fccvrybkas │
├──────────┼─────┼────────────────────────────────────────────────────────────────────┼────────────────┤
│30-08-2024│02:16│Order number 1045534719 / Nice savings with this offer... │cherylebowmiq │
├──────────┼─────┼────────────────────────────────────────────────────────────────────┼────────────────┤
│30-08-2024│02:03│TREND ALERT: Discover one of a kind items │k.a6 │
├──────────┼─────┼────────────────────────────────────────────────────────────────────┼────────────────┤
│30-08-2024│00:12│WARNING! This is not a joke! Carry on reading or you will be cursed!│mark0847 │
├──────────┼─────┼────────────────────────────────────────────────────────────────────┼────────────────┤
│29-08-2024│23:28│I know your password T.J Eckleburg │johnsmith │
├──────────┼─────┼────────────────────────────────────────────────────────────────────┼────────────────┤
│29-08-2024│23:17│Whats with all the spam?? │fluxLinker │
├──────────┼─────┼────────────────────────────────────────────────────────────────────┼────────────────┤
│29-08-2024│22:36│HACKED BY APT1 // FIX YOUR SITE │YouKnowWho │
├──────────┼─────┼────────────────────────────────────────────────────────────────────┼────────────────┤
│01-01-1970│00:00│XXXXXXXXXX │XXXXXXXXXX │
├──────────┼─────┼────────────────────────────────────────────────────────────────────┼────────────────┤
│01-01-1970│00:00│XXXXXXXXXX │XXXXXXXXXX │
├──────────┼─────┼────────────────────────────────────────────────────────────────────┼────────────────┤
│01-01-1970│00:00│XXXXXXXXXX │XXXXXXXXXX │
├──────────┼─────┼────────────────────────────────────────────────────────────────────┼────────────────┤
│01-01-1970│00:00│XXXXXXXXXX │XXXXXXXXXX │
├──────────┼─────┼────────────────────────────────────────────────────────────────────┼────────────────┤
│01-01-1970│00:00│XXXXXXXXXX │XXXXXXXXXX │
├──────────┼─────┼────────────────────────────────────────────────────────────────────┼────────────────┤
│01-01-1970│00:00│XXXXXXXXXX │XXXXXXXXXX │
├──────────┼─────┼────────────────────────────────────────────────────────────────────┼────────────────┤
│01-01-1970│00:00│XXXXXXXXXX │XXXXXXXXXX │
├──────────┼─────┼────────────────────────────────────────────────────────────────────┼────────────────┤
│01-01-1970│00:00│XXXXXXXXXX │XXXXXXXXXX │
├──────────┼─────┼────────────────────────────────────────────────────────────────────┼────────────────┤
│29-08-2024│21:38│Nordtel OC3 Express │corporateRaider │
└──────────┴─────┴────────────────────────────────────────────────────────────────────┴────────────────┘
Not this again, you need to get rid of all the spam before the situation gets out of hand. You first remove all duplicate posts using the monadic unique ∪ function.
∪ POSTS
┌──────────┬─────┬────────────────────────────────────────────────────────────────────┬────────────────┐
│30-08-2024│18:52│Why does DVB-C use QAM instead of OFDM? │frequencySniffer│
├──────────┼─────┼────────────────────────────────────────────────────────────────────┼────────────────┤
│30-08-2024│10:34│DAC in QPSK modulation │radioComputer │
├──────────┼─────┼────────────────────────────────────────────────────────────────────┼────────────────┤
│30-08-2024│02:30│We’ve Received Your Payment │fccvrybkas │
├──────────┼─────┼────────────────────────────────────────────────────────────────────┼────────────────┤
│30-08-2024│02:16│Order number 1045534719 / Nice savings with this offer... │cherylebowmiq │
├──────────┼─────┼────────────────────────────────────────────────────────────────────┼────────────────┤
│30-08-2024│02:03│TREND ALERT: Discover one of a kind items │k.a6 │
├──────────┼─────┼────────────────────────────────────────────────────────────────────┼────────────────┤
│30-08-2024│00:12│WARNING! This is not a joke! Carry on reading or you will be cursed!│mark0847 │
├──────────┼─────┼────────────────────────────────────────────────────────────────────┼────────────────┤
│29-08-2024│23:28│I know your password T.J Eckleburg │johnsmith │
├──────────┼─────┼────────────────────────────────────────────────────────────────────┼────────────────┤
│29-08-2024│23:17│Whats with all the spam?? │fluxLinker │
├──────────┼─────┼────────────────────────────────────────────────────────────────────┼────────────────┤
│29-08-2024│22:36│HACKED BY APT1 // FIX YOUR SITE │YouKnowWho │
├──────────┼─────┼────────────────────────────────────────────────────────────────────┼────────────────┤
│01-01-1970│00:00│XXXXXXXXXX │XXXXXXXXXX │
├──────────┼─────┼────────────────────────────────────────────────────────────────────┼────────────────┤
│29-08-2024│21:38│Nordtel OC3 Express │corporateRaider │
└──────────┴─────┴────────────────────────────────────────────────────────────────────┴────────────────┘
The duplicate XXXXXXXXXX spam posts have been removed. Next is removing specific posts using the set difference ~ function. First, listing all the indices of the post then removing the specific spam posts.
⍴ POSTS
11 4
⍳11
1 2 3 4 5 6 7 8 9 10 11
(⍳11) ~ 3 4 5 6 7 8 9 10
1 2 11
POSTS ← POSTS[(⍳13) ~ 3 4 5 6 7 8 9 10]
POSTS
┌──────────┬─────┬───────────────────────────────────────┬────────────────┐
│30-08-2024│18:52│Why does DVB-C use QAM instead of OFDM?│frequencySniffer│
├──────────┼─────┼───────────────────────────────────────┼────────────────┤
│30-08-2024│10:34│DAC in QPSK modulation │radioComputer │
├──────────┼─────┼───────────────────────────────────────┼────────────────┤
│29-08-2024│21:38│Nordtel OC3 Express │corporateRaider │
└──────────┴─────┴───────────────────────────────────────┴────────────────┘
That's better! Let's take a closer look at some other set functions, too.
Take the following vectors of emoticons
HAPPY_EMOTICONS ← ':)' ':-)' ':D' ':]' ':o)' '8)' ':3' 'c:' ':x'
SAD_EMOTICONS ← '):' ':c' ']:' ')-:' 'D:' '>:(' ':/' ':x' ':|'
The symbol ∪ acts dyadically as Set Union.
HAPPY_EMOTICONS ∪ SAD_EMOTICONS
:) :-) :D :] :o) 8) :3 c: :x ): :c ]: )-: D: >:( :/ :|
Notice how the element :x only appears once, where it would appear twice if we were to use the dyadic catenate , operator introduced in Chapter 2.
HAPPY_EMOTICONS , SAD_EMOTICONS
:) :-) :D :] :o) 8) :3 c: :x ): :c ]: )-: D: :x >:( :/ :|
Similarly, the symbol ∩ acts dyadically as the Set Intersection operation.
HAPPY_EMOTICONS ∩ SAD_EMOTICONS
:x
The symbol ~, which monadically refers to boolean NOT, is dyadically the Set Difference operation.
HAPPY_EMOTICONS ~ SAD_EMOTICONS
:) :-) :D :] :o) 8) :3 c:
⍝ Notice the ':x' emoticon is gone
Monadically, the symbol ∪ acts as the Unique operator, removing duplicate entries in a vector.
∪ HAPPY_EMOTICONS , SAD_EMOTICONS
:) :-) :D :] :o) 8) :3 c: :x ): :c ]: )-: D: >:( :/ :|
⍝ Notice the ':x' emoticon only appears once