nd methods of proof Flashcards","topicId":"187b7f27-807a-40aa-bfcb-044c67c27c35","topicSlug":"structure-and-methods-of-proof","cards":[{"id":"725ebc11-2b3f-472b-a441-2494c7f24cee","cardType":"cloze","front":"A proof proceeds from assumptions through justified logical steps to the required [ ].","back":"A proof proceeds from assumptions through justified logical steps to the required conclusion.","points":null,"mcqOptions":null},{"id":"f55de638-f3ff-4da4-aef9-585cddb1ece4","cardType":"basic","front":"How should an arbitrary even integer be represented in a proof?","back":"As $2k$, where $k$ is an integer.","points":1,"mcqOptions":null},{"id":"9af6c066-f669-4e18-87db-3c8097c6e9fc","cardType":"basic","front":"How should an arbitrary odd integer be represented in a proof?","back":"As $2k+1$, where $k$ is an integer.","points":1,"mcqOptions":null},{"id":"1c623b14-0a98-44df-85e1-07ed30d09afa","cardType":"basic","front":"Why must a proof state that $k$ is an integer in $2k+1$?","back":"It ensures $2k+1$ represents an odd integer.","points":2,"mcqOptions":null},{"id":"932b6dec-644a-430a-be18-6b02bd4b7cf4","cardType":"basic","front":"Which proof method uses definitions and algebra to reach a conclusion directly?","back":"Proof by deduction.","points":1,"mcqOptions":null},{"id":"e477a96f-16c1-4221-822c-333e54e536f1","cardType":"basic","front":"Why must the chosen integer be arbitrary in a deductive proof?","back":"So the argument applies to every allowed integer.","points":2,"mcqOptions":null},{"id":"a877e217-2e2a-4096-8186-a10b5c0c7e1b","cardType":"basic","front":"After obtaining $n^2=2(2k^2+2k)+1$, why is $n^2$ odd?","back":"It has the form $2m+1$ for an integer $m$.","points":2,"mcqOptions":null},{"id":"95c4b1ae-262e-45a4-833f-5d887bbb5940","cardType":"cloze","front":"To prove divisibility by $d$, rewrite the expression in the form $\\text{[...]}$.","back":"To prove divisibility by $d$, rewrite the expression in the form $\\boxed{d\\times\\text{integer}}$.","points":null,"mcqOptions":null},{"id":"33fcb1b0-8fa9-4c75-9415-3e81239b50e6","cardType":"basic","front":"How does $8(k+1)$ prove that the difference between consecutive odd squares is divisible by 8?","back":"It is 8 multiplied by an integer.","points":2,"mcqOptions":null},{"id":"4501603e-df0e-4ed7-8437-1e7c8ea7d486","cardType":"basic","front":"When is proof by exhaustion valid?","back":"When a finite set of cases covers every possibility.","points":2,"mcqOptions":null},{"id":"5b0c5b16-94d7-4d68-99f7-80c8f584df8d","cardType":"cloze","front":"The possible remainders when dividing an integer by $m$ are $\\text{[...]}$.","back":"The possible remainders when dividing an integer by $m$ are $\\boxed{0,1,\\ldots,m-1}$.","points":null,"mcqOptions":null},{"id":"80e5ba47-a13a-4c5f-879a-0c8832de4425","cardType":"basic","front":"What remainders can an integer square leave when divided by 4?","back":"Only 0 or 1.","points":1,"mcqOptions":null},{"id":"bd70a9c1-319a-456b-9ea6-a8b5a0f7642d","cardType":"basic","front":"Why do the forms $4k$, $4k+1$, $4k+2$ and $4k+3$ exhaust all integers?","back":"They represent every possible remainder modulo 4.","points":2,"mcqOptions":null},{"id":"593e277f-39a0-4a9e-80b5-59d14371c103","cardType":"basic","front":"What is sufficient to disprove a universal statement?","back":"One allowed counterexample.","points":1,"mcqOptions":null},{"id":"27db50e6-6b12-47d2-ad96-50a84692512e","cardType":"basic","front":"How does $p=2$ disprove “if $p$ is prime, then $p^2+2$ is prime”?","back":"$$2$ is prime but $2^2+2=6$ is not prime.","points":2,"mcqOptions":null},{"id":"c8051651-2240-4510-ab9e-8836dff686df","cardType":"cloze","front":"A proof by contradiction begins by assuming the [ &nbs

1.4.1 Structure and methods of proof Flashcards

1.4.1 Structure and methods of proof Flashcards