Alg1231W10T2 - The inclusion/exclusion principle

Friday, 28 September 2018

12:10 PM

For two sets A and B the Inclusion-Exclusion principle states that IAU BI = + - IAnBl. Let A be the set of MATHIA students and B be the set of students who live on campus. Suppose that there are 230 students enrolled in MATHIA this year and that 80 Of these MATHIA students live on campus. Suppose also that a total Of 570 students live on campus. In set notation this means IAI = 570 Thus the total number Of students who are either enrolled in MATHIA or living on campus (or ågth, as that is what our usage Of the word 'or' implies) is 720

 

Let A be the set of primes less than 100 and let B be the set of natural numbers less than 100 which are 1 more than a square (such as 1, 2, 5, 10, etc. There are 25 primes less than 100, and there are 10 natural numbers less than 100 which are 1 more than a square (such 1, 2, 5, 10, etc). Together there are 31 natural numbers less than 100 which are either prime or one more than a square. In set notation this means IBI — IA U Bl = How many natural numbers between 1 and 100 prime and one more than a square? 4 Challenge: What are these primes? Answer as a set: {2, 5, 17, 37} By the way, are there an infinite number Of primes that are also one more than a square? This is a famous unsolved problem Of number theory! Recall: the set {1,2,3} in Maple notation is {1 , 2 , 3}.

 

primes : 449, 457, 461, 463, > primes seq(ithprime (i) , 1=1. .100) } ; = {2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251 257, 263, 269, 271, 277, 281, 283, 293, 307, 311, 313, 317, 331, 337, 347, 349 353, 359, 367, 373, 379, 383, 389, 397, 479, 487, 503, 509, 521, 523, 467 491, 401, 499, 409, 419, 421, 431, 433, 439, 443 541 } > + 1, 1=1. . sqrt (100) ) } ; {2, 5, 10, 17, 26, 37, 50, 65, 82, 101} squareplusone > primes intersect squareplusone {2, 5, 17, 37, 101}

 

Of 240 MATHIS students, 44 like poetry, 78 like cooking and 172 like football. Also 13 like both poetry and cooking, 32 like both poetry and football, and 45 like both cooking and football. Finally 7 like all three. How many students don't like any of the three? 29

 

What about yourself? Which of these hobbies/activities do you enjoy? poetry Lj cooking D football none of the above. Note: the Inclusion-Exclusion Principle applied to three sets is IPUCIJFI = + + IFI - - - lcm FI + IPncnFl.

 

b a e C Of 240 (mythical) MATHIC students, 49 like swimming, 106 like music and 165 like texting. Furthermore 37 like both swimming and music, and 27 like both swimming and texting, and 56 like both music and texting. Finally, 21 like all three activities. Let S be the set of students who like swimming, let M be the set of students who like music, and let T be the set of students who like texting. We can represent the situation by placing the numbers on the diagram as shown. Clearly g = IS n M n Tl = 21, and f = 35 since g + f = 1M n Tl = 56 Complete the table below.

 

= 21 d = 16

 

75 10 15 60 5 2 100 If we start with a Venn diagram showing which Of 280 (fictional) MAT HID students like archery (A), badminton (B) and cricket (C) then: i) the number that like both archery and badminton is 20 ii) the number that like both archery and cricket is 15 iii) the number that like badminton and cricket but not archery is 2

 

iv) the number that like only badminton or only cricket is 160 v) the number that like none of these is 13

 

 

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