of mathematical proof Flashcards","topicId":"73ba9497-10da-4a40-8815-6696e54c7ae3","topicSlug":"structure-of-mathematical-proof","cards":[{"id":"b7c8c898-f9d7-4b1c-95bc-dfc20441f4cf","cardType":"cloze","front":"A proof forms the chain assumptions $\\Rightarrow$ [ ] $\\Rightarrow$ conclusion.","back":"A proof forms the chain assumptions $\\Rightarrow$ justified deductions $\\Rightarrow$ conclusion.","points":null,"mcqOptions":null},{"id":"f06bd000-60d8-4c49-a413-6691be13f867","cardType":"basic","front":"What must be true of a proof's conclusion?","back":"It must follow in every case covered by the assumptions.","points":1,"mcqOptions":null},{"id":"1279eba8-85d3-44c1-b146-8516ac03b34a","cardType":"basic","front":"How should an even integer $n$ be represented in a proof?","back":"$$n=2k$, where $k$ is an integer.","points":1,"mcqOptions":null},{"id":"befae858-62ce-4929-9cbc-07a443cd31f6","cardType":"basic","front":"How should an odd integer $n$ be represented in a proof?","back":"$$n=2k+1$, where $k$ is an integer.","points":1,"mcqOptions":null},{"id":"e237f759-8c89-4518-ac17-d7aaca19be1b","cardType":"cloze","front":"An integer $a$ is a multiple of $b$ if $\\text{[...]}$ for some integer $k$.","back":"An integer $a$ is a multiple of $b$ if $\\boxed{a=bk}$ for some integer $k$.","points":null,"mcqOptions":null},{"id":"32b8ca49-67d2-40de-b58b-d52f36185b9b","cardType":"cloze","front":"If $P\\Rightarrow Q$ and $P$ is true, you may conclude $\\text{[...]}$.","back":"If $P\\Rightarrow Q$ and $P$ is true, you may conclude $\\boxed{Q}$.","points":null,"mcqOptions":null},{"id":"220a108f-6d60-4b4c-bc91-a14c9eeaef27","cardType":"basic","front":"Why can $P$ not be concluded from $Q$ when only $P\\Rightarrow Q$ is known?","back":"The converse $Q\\Rightarrow P$ may be false.","points":2,"mcqOptions":null},{"id":"ee0f9c2d-11bd-40f1-b18f-caae315ad13d","cardType":"cloze","front":"Successful examples provide [ ], not proof of a general statement.","back":"Successful examples provide evidence, not proof of a general statement.","points":null,"mcqOptions":null},{"id":"6a19ecfd-a103-4f05-92a5-95b3fa8920f1","cardType":"basic","front":"What is sufficient to disprove a universal statement?","back":"A single counterexample.","points":1,"mcqOptions":null},{"id":"d09a5d1d-50f8-4c8c-acff-56e77aefcb8e","cardType":"basic","front":"Why is simplifying an odd square to $2(2k^2+2k)+1$ not quite a complete proof?","back":"It must be linked to the definition of an odd integer.","points":2,"mcqOptions":null},{"id":"f14c4d07-f660-4a76-96e4-1da0db215113","cardType":"basic","front":"How does algebra show that the square of $n=2k+1$ is odd?","back":"$$n^2=2(2k^2+2k)+1$, which has the form $2m+1$.","points":2,"mcqOptions":null},{"id":"d9cdc818-af1e-401c-918a-d09379a737e3","cardType":"basic","front":"How should two arbitrary odd integers be represented?","back":"$$2a+1$ and $2b+1$, where $a,b$ are integers.","points":1,"mcqOptions":null},{"id":"68607c22-d2c4-4604-9df0-96230c3d3fe5","cardType":"basic","front":"What false assumption is made by representing two arbitrary odd integers as $2k+1$ and $2k+1$?","back":"It assumes the two integers are equal.","points":1,"mcqOptions":null},{"id":"cb1f39de-2330-4fca-a6de-11f90f8cdd70","cardType":"basic","front":"How does deduction show that the sum of two odd integers is even?","back":"$$(2a+1)+(2b+1)=2(a+b+1)$, which is twice an integer.","points":2,"mcqOptions":null},{"id":"22990ea1-08cd-404f-a29f-9f9a04346be0","cardType":"cloze","front":"To prove divisibility by $d$, rewrite the expression as

1.1.1 Structure of mathematical proof Flashcards

1.1.1 Structure of mathematical proof Flashcards