f by contradiction (A-level only) Flashcards","topicId":"103b856d-3296-4fff-852d-ef9b3ab679e9","topicSlug":"proof-by-contradiction","cards":[{"id":"47609d90-af9b-4ddd-b5c8-176a8a900380","cardType":"cloze","front":"The negation of a statement $P$ is [ ].","back":"The negation of a statement $P$ is not $P$.","points":null,"mcqOptions":null},{"id":"ad51cf95-0795-4bb8-854a-fa7b60df6fee","cardType":"basic","front":"What is the negation of “Every object has property $P$”?","back":"There exists at least one object without property $P$.","points":1,"mcqOptions":null},{"id":"c1255306-4e2b-4c80-a6d5-2f77fa1ce377","cardType":"basic","front":"What is the negation of “There exists an object with property $P$”?","back":"No object has property $P$.","points":1,"mcqOptions":null},{"id":"e8c72f8c-2e34-4ba6-a5d7-17c23c00a565","cardType":"basic","front":"Negate: “For every integer $n$, $n^2+n$ is even.”","back":"There exists an integer $n$ for which $n^2+n$ is odd.","points":1,"mcqOptions":null},{"id":"19acec0a-7a6d-4e64-a1e8-ced0a45efb7a","cardType":"cloze","front":"If assuming not $P$ produces a contradiction, then [ ].","back":"If assuming not $P$ produces a contradiction, then $P$ must be true.","points":null,"mcqOptions":null},{"id":"0694edad-27b2-4dd3-9e17-ebb0db949adc","cardType":"basic","front":"What must be assumed first in a proof by contradiction?","back":"The negation of the required conclusion.","points":1,"mcqOptions":null},{"id":"dd7c206b-7441-4b68-b9a0-797a61128a70","cardType":"basic","front":"What makes a conclusion a contradiction?","back":"It is logically impossible or conflicts with a known fact.","points":1,"mcqOptions":null},{"id":"fae0429e-b343-409c-a5b0-c6d5d670d38a","cardType":"basic","front":"After reaching a contradiction, what conclusion completes the proof?","back":"The initial assumption is false, so the required statement is true.","points":2,"mcqOptions":null},{"id":"d2f6bcbf-fdf4-487f-9045-f4df3661ae22","cardType":"cloze","front":"A rational number can be expressed as [ ].","back":"A rational number can be expressed as $\\frac{a}{b}$, where $a,b$ are integers and $b\\ne0$.","points":null,"mcqOptions":null},{"id":"6f55d336-a69e-4160-a70c-badda69df197","cardType":"basic","front":"Why must $\\sqrt{2}=\\frac{a}{b}$ be written in lowest terms in the contradiction proof?","back":"So proving $a$ and $b$ share a factor contradicts the initial assumption.","points":2,"mcqOptions":null},{"id":"9d4511e3-cb8e-4d21-8108-99201f754013","cardType":"basic","front":"If $a^2=2b^2$, what follows about $a$?","back":"$$a$ is even.","points":1,"mcqOptions":null},{"id":"d42b3791-e569-4360-b5f2-e7f28ae7948a","cardType":"basic","front":"After writing $a=2k$ in the $\\sqrt{2}$ proof, what equation follows?","back":"$b^2=2k^2$","points":1,"mcqOptions":null},{"id":"764b1fb8-748b-47d8-856a-4e3bb2f90d9b","cardType":"basic","front":"What is the precise contradiction in the proof that $\\sqrt{2}$ is irrational?","back":"$$a$ and $b$ share factor 2 despite being chosen coprime.","points":2,"mcqOptions":null},{"id":"1302f8e1-5fe3-4656-969d-2ad8bfa018d9","cardType":"cloze","front":"If $n=2k+1$, then $n^2=2(2k^2+2k)+1$, so $n^2$ is [ ].","back":"If $n=2k+1$, then $n^2=2(2k^2+2k)+1$, so $n^2$ is odd.","points":null,"mcqOptions":null},{"id":"06b7df72-e2b8-4a1f-b783-58925d67837e","cardType":"basic","front":"Why does an even value of $n^2$ imply that integer $n$ is even?","back":"If $n$ were odd, its square would be odd,

1.1.3 Proof by contradiction (A-level only) Flashcards

1.1.3 Proof by contradiction (A-level only) Flashcards