Misplaced Pages

Revision as of 19:23, 24 March 2023

An example is, according to Dictionary.com, an "instance illustrating a rule or method, as a mathematical problem proposed for solution." Like most English words, the word example has several other closely related, similar definitions, such as "a pattern or model, as of something to be imitated or avoided"; this sense is typically encountered in the common expression, "set a good example." Good parents must exhibit proper behavior around their children to set a good example for them. If you use profanity around children, the children will likely pick up the profane words and begin using them themselves.

In this article, we will focus on a specific sense of the word. An example, in logic and set theory, is an item that illustrates a rule or method or an item that is not a counterexample. For example (pun intended), the numbers 7 and 10 are examples of the statement, "All prime numbers are odd." Seven is not a counterexample because seven, while prime, is indeed odd. Ten is not a counterexample because ten, while not odd, is not prime. In this case, the only counterexample to the statement is two because it's the only even prime. In a mathematical pun, it is sometimes called the "oddest prime", using the non-mathematical sense of "odd" as a synonym for "strange."

Examples

• The word "Examples" in the header title is, of course, being used in the sense, "instance illustrating a rule or method, as a mathematical problem proposed for solution."
• Take the statement, "All positive odd integers are either square or prime." At first, the statement appears to work well. One and nine are squares, as 1 = 1, 3 = 9. The numbers 3, 5, 7, 11, and 13 are all prime. None of those numbers have positive divisors other than one and themselves. The numbers 1, 3, 5, 7, 9, 11, and 13 are all examples of the statement, as none of them are counterexamples. The smallest counterexample to the statement is 15, as 15 is not a perfect square (the square root of 15 is about 3.872983, which is not an integer), and 15 = 3 × 5, so it is not prime. The strong law of small numbers says that one should not assume that a statement always holds just because it holds so far. Some conjectures, such as the comparison between the prime-counting function and the logarithmic integral (see Skewes's number for more information) and Pólya conjecture, have been disproved using large counterexamples. Pólya conjecture is discussed in the next bullet point.
• For any natural number n, let f(n) be the number of prime factors of n, respecting multiplicity of prime factors. For example, the number 6 has two prime factors because 6 = 2 × 3, and 30 has three prime factors because 30 = 2 × 3 × 5. Repeated prime factors are counted repeatedly, so we say that 48 has five prime factors because 48 = 2 × 2 × 2 × 2 × 3. Thus, f(6) = 2, f(30) = 3, and f(48) = 5. Pólya conjecture states that for any natural number n, at least 50% of the natural numbers less than or equal to n have an odd number of factors.
  • The conjecture holds for all small integers. For example, the numbers between 1 and 20, inclusive, with an odd number of prime factors are 2, 3, 5, 7, 8, 11, 12, 13, 17, 18, 19, and 20. Pólya conjecture holds for n = 20 because that's 12 of the 20 numbers, and 12/20 = 60% > 50%. Pólya conjecture also holds for n = 50, twenty-eight of the first fifty positive integers have an odd number of divisors: 28/50 = 56%. Pólya conjecture even holds for n = 100, with 51 of the first 100 natural numbers having an odd number of divisors: 51/100 = 51%. What about n = 1000? Same thing. Of the first 1000 natural numbers, 507 have an odd number of divisors: 507/1000 = 50.7%.
  • So, does the conjecture always hold? Nope! The conjecture holds for all integers up to 900,000,000. In 1958, mathematician Colin Brian Haselgrove disproved the conjecture, showing there was a counterexample somewhere around 1.845 × 10. Two years later, a vastly smaller, explicit counterexample was found, that being n = 906,180,359. In 1980, Minoru Tanaka found the minimal counterexample to be n = 906,150,257. The size of the smallest counterexample is often used to demonstrate the fact that a conjecture can be true for many cases and still fail to hold in general, providing an illustration of the strong law of small numbers.
• Goldbach's conjecture states that all even integers greater than two are the sums of two primes. For instance, you can write 4 as 4 = 2 + 2, and you can write 50 as 50 = 47 + 3. There are six ways to write 100 as the sum of two primes: 3 + 97, 11 + 89, 17 + 83, 29 + 71, 41 + 59, and 47 + 53. Goldbach's conjecture states that every even integer greater than two has at least one such partition. This conjecture has been shown to hold for all even integers up to 4 × 10, but it remains unproven despite considerable effort. In other words, no counterexamples are known. However, heuristics suggest that the conjecture is very likely to be true, as large even integers should, heuristically speaking, have not just one, but many partitions into the sums of two primes. It is known that for sufficiently large n, the density of prime numbers among the positive integers less than or equal to n is approximately ${\displaystyle 1/\ln n}$, this is known as the prime number theorem, first proved by Jacques Hadamard in 1896.
• Riemann hypothesis states that all zeroes of the Riemann zeta function are located either at the negative even integers or the complex numbers with real part ⁠1/2⁠. All zeroes found so far have been found in one of those two categories; as of 2023, no counterexamples have been found. Riemann hypothesis remains unresolved.
• Many IP editors are vandals, but not all IP editors are vandals. In fact, some IP editors, such as 130.216.1.16 are incredibly well-behaved and have made lots and lots of useful edits. 76.66.193.90 has made thousands of legitimate contributions to the English Misplaced Pages.
• Not all CL words are insults. Although clumsy, clueless, and clingy are negative, insulting words, there are plenty of counterexamples. Some CL words are specifically used as compliments, such as classic, classy, clever, and clement. The word climactic has a positive denotation and connotation as well, and it is usually used and encountered in discussions and reviews about movies, video games, literature, and other media with plots or storylines. Other CL words, such as clockwise, are completely neutral. It means "in the manner or direction of a clock", with no other secondary meanings. Positive nouns include cleanliness, clairvoyance, and clarity. You can even use classic as a noun to describe something that is of enduring excellence, or even the author of such a work. You can also use cleansing as a compliment; plenty of YouTube videos exist that provide "cleaning frequency music" for relaxation and meditation.
• Is there a stigma against female singers who sing in the key of G? Well, I hope not! Rachel Platten's "Fight Song" has an abysmal rating of 30/100 based on 40 user ratings on Album of the Year. Seriously, why the FUCK do people hate this beautiful song? It is one thing for Ed Sheeran's "Shivers" to get a low rating. "Shivers" is a mediocre song that lacks substance. But a song about being strong, with powerful female vocals, for a song like that to get a low rating? That's another thing entirely. It doesn't help that Todd in the Shadows ranked this song as #2 on his "worst hit songs" list for 2015, behind only Charlie Puth's "Marvin Gaye." Pink's "A Million Dreams" is another beautiful, inspirational song that the world is too blind to appreciate, giving it only 38/100 in 7 user ratings. However, not all female songs in G get low ratings. Miley Cyrus' Midnight Sky is a universal favorite, getting 83/100 based on 192 user ratings. Lizzo's "2 Be Loved" has 75/100 based on 38 user ratings. Over the past few weeks, several new additions to the radio are also counterexamples. I guess women in the music industry are learning from their mistakes and being careful not to create another anticlimactic disaster such as "Fight Song" and "A Million Dreams." All winter, I have had lingering cognitive dissonance over my conjecture that when it comes to the key of G, the sex preference switches. Normally, songs with female vocalists are more likely to be well-received (and less likely to get low ratings) than songs with male vocalists. Why, then, does the sex preference switch to male over female when it comes to key of G songs? Todd in the Shadows may have hated "Fight Song", but he loved "Cake by the Ocean." Even Justin Bieber has a song called "Company" which has a respectable 74/100 rating even though most Justin Bieber songs are either lowly rated or decent at best in terms of Album of the Year ratings. Well, it seems like the Music Gods finally answered my prayers, heard my pleas, and felt sympathy for me, seeing as how there are now plenty of new additions that are counterexamples to my conjecture. I would have been okay with just one new counterexample, but now we are actually getting two or three at once? Raye's "Escapism" (87/100), SZA's "Nobody Gets Me" (81/100), and Beyonce's "Cuff It" (89/100). Excellent key of G songs come in threes. Excellent things come in threes.
• While we're on the subject of music, not all songs by male vocalists get low ratings. Most or all songs by The Weeknd, a male singer, are rated above 80/100. "Can't Feel My Face", "Starboy", "Save Your Tears", all very well-received songs! As illustrated by "Company" in the previous bullet, not all Justin Bieber songs get low ratings even though many of them do. Another relatively young boy with high ratings is Harry Styles, who was only 23 years old when he released "Sign of the Times", which currently enjoys a rating of 84/100 based on 170 user ratings.
• Not all boys are Justin Bieber haters. Some of them, in fact, love him and his music. These boys are thus counterexamples to the claim that "All boys hate Justin Bieber" or even "All boys dislike Justin Bieber."
• Not all franchises are falling in quality. While Pokémon games and Maroon 5 songs are, indeed, getting lower ratings over time, Super Smash games are just as universally beloved in the latest title ("Super Smash Bros Ultimate has 8.6/10) as in the classics from the start (Melee has 9.1/10). And no, the difference of 0.5 points is miniscule when the numbers are 8.6 and 9.1 because both are high enough to be classified as "universal acclaim" or "universal praise." Anyone who has played any game of the Super Smash Bros. series for at least a year has the intuition and clairvoyance to foresee that Super Smash 5, whenever it may be released, will also get above an 8/10, or at the very least, a 7/10, and be excellent just like the previous titles.
• Justin Bieber may be banned from visiting China, but not all countries have blacklisted or banned him. There are still plenty of countries where he is allowed to tour.
• Although "Yummy", "Lonely", and "Peaches" got entries on Todd in the Shadows' "worst hit songs" year-end lists, not all Justin Bieber songs did. "Ghost" and "Anyone" were absent from Todd's "worst hit songs" lists. Unfortunately, Todd is currently a Justin Bieber hater, and no Justin Bieber song, as of March 2023, has made it on any of Todd's "best hit songs" lists yet.
• Not all Pokémon are hostile to humans. Although you wouldn't want to come face-to-face with a Sharpedo or an Aggron in real life, you would certainly want a Blissey in real life because Blissey is a kindhearted Pokémon that nurses sick Pokémon and people back to health. Blissey has eggs that are filled with happiness. Eating even one little bite will bring a smile to anyone.
• Not all boys have bad tempers.
• Most boys are not school shooters. Although boys are far more likely to be mass shooters than girls (with 122 of the 126 mass shooters since 1949 being male), all this implies is that "most shooters are male." Note that "most X are Y" does not imply "most Y are X", and it is a logical fallacy to assert that the former implies the latter. In this case, "most shooters are boys" does not imply "most boys are shooters." The latter is clearly false; you undoubtedly know lots of boys and men in your life who are sweethearts who would never hurt a fly. Only a tiny fraction of people will ever become shooters. However, since shooting often causes death, even one shooter is one too many, so if you have cause to believe that someone could be dangerous, report the person immediately!
• Not all changes to YouTube in the past five years have been bad. While the November 2021 removal of dislike counts was universally hated, there are some positive changes, too. In the spring of 2022, YouTube added the "most replayed" feature that allows viewers to see when the most replayed part of a video is. The most replayed part of Lady Gaga's "Perfect Illusion" video is 1:49, when the song changes key.
• Not all "overplayed" or "frequently played" songs on the radio are disliked by the audience. Not all charting songs are bad. Way too often in pop music, some people complain that a song sucks and does not deserve its popularity. When "Yummy" won the Grammys, people were upset because "Yummy" is a crappy song. What kind of drugs was Justin Bieber and his teammates on when they decided that making a song of saying the word "yummy" over and over again was a good song. Don't get me wrong. I want you to know that I am a huge Justin Bieber fan, and I love almost all his songs. "One Time", "Love Yourself", "Intentions", those are all amazing tracks! "Yummy" is the only exception; I don't like that song. Adele's "Someone Like You" was a universal favorite that was #1 on Billboard Hot 100 in 2011. Despite the existence of lame, crappy songs that charted and won awards, claims that pop music is "falling apart" are not accurate. On the year-end Billboard Hot 100 list for 2021, the top 3 were Dua Lipa's "Levitating", The Weeknd's "Save Your Tears", and The Weeknd's "Blinding Lights", all of which are incredibly well-received, getting above 80/100 on Album of the Year. Another beloved favorite, "Leave the Door Open", finished 2021 at #7.
• Not all children songs are bad. While "Baby Shark" is terrible, there are plenty of kid songs that are amazing to listen to at all ages. Examples (counterexamples to the claim that "all kid songs are bad") include "Let It Go" from Frozen, Cocomelon's Beach Song , and "Mother Earth, Father Time" from Charlotte's Web (1973). Also, Toy Story movies are great for all ages, and most viewers also enjoy the song "When She Loved Me", sung by Jessie at around the 50-minute mark of Toy Story 2. If you need good music to help you fall asleep, try Celtic Dreamland. There are many social stigmas that need to stop; in a perfect world, there would be no social stigma against adults who enjoy children's songs, children's games, and children's movies.

References

1. "Example Definition & Meaning". March 23, 2023.
2. "House Cleanse 8 Hours House Cleansing Frequency Music". August 7, 2017.
3. "Rachel Platten - Fight Song". February 19, 2020.
4. "Top 10 Most Hostile Hoenn Pokémon". November 23, 2014.
5. "Lady Gaga - Perfect Illusion". September 20, 2016.
6. "Beach Song". July 9, 2019.
7. "Dream Angus". February 19, 2021.

• Mathematical terminology
• Logic
• Interpretation (philosophy)
• Methods of proof