mathematical proof exercises

JdK146fc3qQlKxF7cSemSkgLenMI1frWh2OKrdA8x+aNR/Jr/FMl8n6dm02bUY3EKLArxozpH6dC Prove P(0). PROCESS 100.000000 50.000000 0.000000 V8r6T5X0SHRtKWQWkLO4aaRpZGeVzJI7u25LMxOKptirsVdiqHvLZ5YJFgKRzSAAyMnLYdO46dvD PROCESS 3/bRuP8AiEeKpB/zmH6v+GtBrx9L64/Hry5ek1fopTIeri6cNfG+v6FQ35rw6lc/84weV5bdg1vb C8kGoxksbOKuyJOzCnRfs/spiqbfmNY6nZeWYdT1n0tQ1nUr+ysgsyC403TEu51jLxW0oEUhirQS q Exercises Prove each of the following. C=0 M=10 Y=95 K=0 0.000000 90.000000 ibgoO5RfTlJqQOleAGZcdJLDPFGNnHE130OCfPy+kBF3aYmGEliUUlxRjQVIpTfNqccTew3YIRbG d2N0UVwA6CROSNsfBh/HFWHfkx+YmseZtK1a08ziKDzDodzwvhGPTjNvKvOGUAnZWAbfwFe+Kqf5 ��-�ЧR��s�TI�}Uͩ�}��f6�� I#��b侴�v>~u�SR� 6�7*"T�/8�P�w4��=��\L��C 60.000004 PROCESS dFt/KtzDql5dHWW1lL1r6W4aQ+ot0Ihw71VfA16nFXsH5QeU9a8p+RrTQNYuLe5urOWcK1pJJJEs 10.000002 0.000000 e2OgkhTqHP8AdhmYIFburcjShFcVQWn/AJsflxqPmAeX7HX7W41ZmKJAhYq7ivwpLT0nbborE4qq 95.000000 6bUzDkDjn6BUeZ+Hd91fJlzRkUqSoHQ1U9Kgjpt0OZkJiQscmK7Jq7FXYqp3KF4HUAEkbVNPxyvN PROCESS CMYK In the world of numbers we say: Step 1. Then, when I release them to practice on their own, they often stare at the page. 100.000000 50.000000 CMYK E2o3zmtOjMq8R/sa4FS7y5+Yf5rflh+YNl5W883smpaVdvEsj3EjXP7qZuC3FvcSUkIRuqt4EUBp 19.999998 112 P13mcD9lYQf9kVHfFVPzD5j8z6d+TR80w3ynXIdOh1GR2hQws8iI7x+nQEJ8R4718ScVXaxqvnof 6.2 A More General Principle of Mathematical Induction. 1OT0ZXWFlaW1xdXl9WZ2hpamtsbW5vY3R1dnd4eXp7fH1+f3OEhYaHiImKi4yNjo+Ck5SVlpeYmZ DzTo8uka7ZpeWMu/BqhlcdHRxRkYV6g4qwWz/wCcefJ8U0H17Vdb1axtXWS20u/v2e1jKGqcUjSM PROCESS CMYK For more information contact us at info@libretexts.org or check out our status page at https://status.libretexts.org. End of proof: Therefore $n$ can be written as the sum of consecutive integers. rF5ZLTTohBBJOVaQqvdiqotfkoxVg3n3/nHr8vfOF9JqU0U2l6pMeU93YMqCVj+1LG6uhPiwAY9z CMYK Unless otherwise noted, LibreTexts content is licensed by CC BY-NC-SA 3.0. 8ZPBSevEEniPYYqj8VdirsVdiqE1G7uLWNZo4fVhU1uSD8SxjqVHcjrmFrdTPDETjHiiPq7xHvA6 C=40 M=65 Y=90 K=35 $ \newcommand{\lt}{<}$ C=75 M=0 Y=75 K=0 Proof. +cv4UP5eaVMftpq8aL8ntrgn/iAxVKvON/c2/wDziVpSxuR9ZtrGCRqmvp+sG418PgA+W2BWMflL converted Tmu1R8xQ4qk50n/nIbTYo78a9pWuSBla60d7QQLw2DJBMpRmIFSOfH+GKsq80/mj5A8q3iWWva1B 35.000004 PROCESS Show it is true for first case, usually n=1; Step 2. 0.000000 module we introduce the basic structures involved in a mathematical proof. 0.000000 PROCESS Let $a$ and $b$ be integers. Prove using mathematical induction that for all n 1, 1+4+7+ +(3n 2) = n(3n 1) 2: Solution. C=75 M=0 Y=100 K=0 qv6J1fXre21AbSW4DyGM0rSUxK6xmn85GKsqtLu1vLWK7tJkuLWdBJBPEwdHRhVWVlqCCO4xVq7s 0+PzhJSktMTU5PRldYWVpbXF1eX1RlZmdoaWprbG1ub2R1dnd4eXp7fH1+f3OEhYaHiImKi4yNjo CMYK 0.000000 0.000000 0.000000 The statement P1 says that 1 = 1(3 1) 2; which is true. We know a number can be represented by n, so the next number would be (n+1). 0.000000 =�HM�1��w��n � (Q@�n��$5&+��()+��:��uV_�Q�� CJ�u �A)��UOD�*�v�[�oKQsK��ʓ�y�y� �+�Mm�B�֦\�rv��s��- ��;p�Ax�EQ@Rێ�rRڒ��o�3H�\u i� }J�G��'n|�w�[iy2B� �9 ��yV�ΜF{�� V酧��PZ��3��2G��s^��ǻ!��e�AP�3��x�V}�R ACj5pi0nRmmlWOze6uYSHVqO3CKGRdz/AJWBWYfnTo3mvRv+cfrLT/NeoLqmtxaohuL1JJJVZWaZ 100.000000 69.999702 3. 100.000000 R�3�"� Look at the truth value of $Q$ in each of the rows that have $P \vee Q$ and $\neg P$ true. Then $n = 2k$ for some integer \(k\text{. This will happen in most mathematical proofs. PROCESS Three cards that are either all the same suit or all different suits. WPlbU9J/LLyp518raZLB5v8ALaomraetvJDPeWzMUuIJYiqs7bhg1CaVI3pir0fX9V0XzBfeW4bm CMYK 8oPLmo+bH8zw32paXqFwYm1CLTrowQXZgoE+sR8W5CgAoCK/fiqe+VPJ2keV4tSi0wyldVv59Uuv MbU4DOBECIyl1Iv9Xw7kgoLTIbtIrgTyuyKiQo0xNC6BubL9k8SWoPGma/QYssRPjlIgRERxd4vi 85.000000 75.000000 yf5u8wa35VvdEFgsHl/UE1Gf65PNG8vEgGJVjgmAqtfiJ+jFWb32n2WpWEtjqVtFdWlwnC4tZVEs Then say how the proof starts and how it ends. PROCESS If we only had three different values, that would be only 9 dice, so there must be 4 different values, giving 4 dice that are all different. 3. We can have 9 dice without any four matching or any four being all different: three 1's, three 2's, three 3's. BIPRSnJipJLKTtxI+GlMxbnmuB9I+/3dOX9ieSLmkSCIfCWGyhVFTvmXkmIR5fJCCiWqmyZ+MiN+ False AQBIAAAAAQAB/+4ADkFkb2JlAGTAAAAAAf/bAIQABgQEBAUEBgUFBgkGBQYJCwgGBggLDAoKCwoK ��ZZ��МL��#q�pMB��\m�zHa��0Ce��r��pqAk]/��M�\.sY���2Ol��[E� �fZ�5. qdl9anvzFZgaXKI6Cb0WrLcV5xpx+yK1xVOfM/n+XRfOmheXFtIpU1gqHuXno6FnKhVgRZHOys3I $ \def\R{\mathbb R}$ 85.000000 PROCESS $ \newcommand{\va}[1]{\vtx{above}{#1}}$ PROCESS PROCESS Bonus points for filling in the middle. kbBqAfEDiqM/OLzxqvlDyo19o1uLrVS6usJAIW2gIkupWr+yIxxr2LDFWRHVP0x5UXVdFuvSF7aL PROCESS 6. 9.999103 K7ubVS0j+ofsxgUCjtiqC8wfln5t1XyP5WigubKy88eUmtmsL9ZJHtpPRRYpQ7+ikipMq8mUR9QB Chapter 5 Supplemental Exercises. 50.000000 0.000000 IjG0np9WljVq+OwPVaFVkH5oaJq1n548oefbG1lv7TQXuLfVrW3QyzJbXcZjM8ca1d/TDsWVQT09 $ \def\iffmodels{\bmodels\models}$ $ \def\circleB{(.5,0) circle (1)}$ It takes practice to learn how to write mathematical proofs; you have to keep trying! This includes im-proving writing techniques, reading comprehension, … PROCESS (Constructive proof.) $ \def\C{\mathbb C}$ py�? C=0 M=100 Y=100 K=0 0.000000 2013-01-13T21:46:07Z 0.000000 0.000000 uuid:5D20892493BFDB11914A8590D31508C8 LqWpK1xc3DGPT9PjIEk8gFTua8UWo5NTb3JAwq8asPzi/wCciPOitdeUPLUMGmsSsNx6VVJ6H/SL PROCESS 10.000002 Lry67pl3JlNcX4nEipH9QADvIwPMLQljSvy7f27TLmzCfEBHweZJu+t/o6d/dvEAfFXs3t5bZJYE wxVOb3zxfWPkTTvMd1Z26XOoG0Bia4ZLSBb2RVSSa4aOqpGkgZzw67e+KoDy3+Zeqa75i0jTYdF9 A flush of five cards (for example, five hearts). Show that if n=k is true then n=k+1 is also true; How to Do it. 508pfn5+XFrF5pvfM01zF6yLNPbX1zcBHevH147hUVlJ23DLXFWSx/mB+dX5vWVtpnk+A6HY28KJ saved Proof that π is irrational. Logic and Proof Exercises. PROCESS C=0 M=35 Y=85 K=0 Clearly state the style of proof you are using. ?�h&vMUb��y�-�$�Y��1��[��;v=�)KV��ݮ`�nq$͌2�v��q�'��^�4��KCi Ju� y �`S�ҷV�^��Z9kN��\�K��eF�(�R Now, So $a^2$ is a multiple of 3. 0.000000 6.3 Proof By Minimum Counterexample. Prove the statement: For all integers $a\text{,}$ $b\text{,}$ and $c\text{,}$ if $a^2 + b^2 = c^2\text{,}$ then $a$ or $b$ is even. Proof: Interesting activities in conjecture and mathematical proof is written by a teacher and all activities have been throughly trialled with students. 4iuFUZ+UVvb335hfmVf6nFHLrMerG0V5AGdLFQyW6itaK8a7+NPbFXnHm2G3svKn526XpiiPQ7bU A standard deck of 52 cards consists of 4 suites (hearts, diamonds, spades and clubs) each containing 13 different values (Ace, 2, 3, …, 10, J, Q, K). Can you still cover the remaining squares with dominoes? In 1-4, write proofs for the given statements, inserting parenthetic remarks to explain the rationale behind each step (as in the examples). Improve their quality of communication in mathematics. PROCESS Simplify the statements below (so negation appears only directly next to predicates). That is, there is an integer $n$ such that $8n$ is even but $n$ is odd. Black Therefore, $\log(7)$ is irrational. Assuming the statement is true, what (if anything) can you conclude if there will be cake? Therefore, a direct application of your result yields, b 2 − 4 a 2 = ( b − 2 a) ( b + 2 a) ≤ ( b − 2 a) + ( b + 2 a) 2 = b. Proof by contradiction. CMYK Red 65.000000 $ \newcommand{\amp}{&}$, $ \newcommand{\hexbox}[3]{ 0.000000 C=0 M=90 Y=85 K=0 <>stream e"]�~��UӾ��KrCr4��,�2E��c��n�j��kr��W=��?��~��j��iwg�Ql:w�(]��.�Ϫ3B�G�w�ЏO��o�B+��f�*��1We�>��^��WmL�sW�>/0 9�=��幍�x8��ϪM�&��ދ�3��s3��G�\4�G�d"#��Db��@B��Kz�:hO� OWOsPGMBTSdvtct6YqnH5a/ld5f/AC90+8sdFuLu4ivZRPK148bsGVeNF9OOIUp4jFVv5jfln5O8 oL8uNB0S/wDzO/Mu6vrC3u7hL6CBJJ4kkKxSQkug5A0VqCo74q8v8z2tlZfl9+bVhpKhNCt/MFot Simplify the following statements (so that negation only appears right before variables). 0.000000 GZKHYq8ksvNvnW61/wDMPTH1y3s4fKK28lhdTW0RQpcQyzf6Tun2RGBVePfFU60fzj5n1z8mR5t9 25.000000 Equivalently, we could just prove the logical negation of the given statement, which is the statement 9x8y: y 2 x. \( \def\circleAlabel{(-1.5,.6) node[above]{$A$}}$ $\newcommand{\twoline}[2]{\begin{pmatrix}#1 \\ #2 \end{pmatrix}}$ V+/FUh/KT8uf8CaDdaa8/wBZkmvJ5IZCxbhaiRvq0QrSlEPNh2dm69cVS1/J3nuH809U82w22k3W CMYK Hint: you should get three T's and one F. It's your birthday, but the cake is a lie. Now, about this sleepless problem. C=0 M=0 Y=0 K=40 $ \newcommand{\f}[1]{\mathfrak #1}$ }\) End of proof: … this is a contradiction, so there are no such integers. 1u20/Vre9iuopLScelHc2/p8Lj4KREmq/F3xV5vD5U81aV+UHmbTdchmurrTY7rQfLMEUMks0tu1 3NfwXETdGjk0uVWU/MHAqZfkZf3f5e/m5rn5canIwtL6RhYvIdjLEDJBINqD17c7+/EYqj/+chtR 80.000000 xLVuNd6MrK6+xwq+avy8/M789NcmvdA8v3tzqupagsf+mXL+p9TiQt6kitJ+7j5cwCzeFAORGBUZ 0.000000 Example: Prove algebraically that the sum of two consecutive numbers is odd. PROCESS $\forall x \exists y (O(x) \wedge \neg E(y))\text{. \( \def\AAnd{\d\bigwedge\mkern-18mu\bigwedge}$ Mathematical induction and a proof. PROCESS CMYK 29.998802 75.000000 Clearly state which statement is $P$ and which is $Q\text{.}$. 0.000000 v5tlvppLdZJIrK0F9F6PpxuIndxEu/8AdryPZRirKPy/0bzDoHkbTdF1JLSTUtLtltYzbzSNBIIh C=0 M=0 Y=0 K=90 7KKAKoXUPICeZ/y5g8q+bI4frSW0UTXFo7SiO4hQKtxEzxwkGorQr0+HcYqlWo/l95oP5Lr5Gspr 90.000000 4. study and an extra exercise in constructing your own arguments. 100.000000 There is a Unwinding Definitions (Getting Started) Constructive Versus Existential Proofs; Counter Examples ; Proof by Exhaustion (Case by Case) B5P0XXvKeqPdQRre2nmK5a1vgo9aNbWeOSOMOOLen8PHhXjQ9MVWyqn5XfmxdTrZS3eh+c4SulJC PROCESS 79.998795 In principle we try to prove things beyond any doubt at all — although in real life people wSNompQahJarEbxbdifTaUHiDXpXg1PliqRfmn+Yl95Xj0vStDs11HzT5hnNrpFrISIlIoGmlpvw PROCESS ��D{@k(J�pr��2�VeuL�[��x� (*H88��9��}{�\��ը �#�|��i)ImK��:�ګ3�͉mfB�KaAm%��BZ 90.000000 CSiINwMvjISFg2EPSPK/l2y8t+XrDQrF5ZLTTohBBJOVaQqvdiqotfkoySvNvPH/ADjL5E80azPr In the following sections, we want to show you how to write mathematical arguments. 2lXC+isxliikPEu/xNuygEV4jq3U5h6OP7g6bIOES4og7Ey3luRyuusucr2SedopLCW1ubFImaS2 There is a number $n$ which is not between any other two numbers. application/postscript }\) But then $a+b = 2k + 2j = 2(k+j)$ which is even. Prove that if n is an odd integer, then n2 is an odd integer. Gödel's completeness theorem and its original proof. PROCESS n + (n+1) Step 2: Expand / simplify the expression as much as possible. PROCESS Fill in the blanks proof (Practice Problems 1 and 2, Exercises 2, 14, and 18) Section 7.3: Isomorphism and planarity. 1.4. 100.000000 $ \def\land{\wedge}$ Make a truth table for the statement $\neg P \wedge (Q \imp P)\text{. Proofs about graphs and trees. Millimeters %�� }$ And we all agree this is true. 39.999996 PROCESS }\), $(P \wedge Q) \wedge (R \wedge \neg R)\text{. ZQT h8�:��$n7��zf�v��Hٵw��7�R,�W�ʊ�/ }$ Note, you will need to prove two “directions” here: the “if” and the “only if” part. For example, consider $n = 3\text{. endobj C=85 M=10 Y=100 K=10 \begin{equation*} 3 = \frac{a^2}{b^2} \end{equation*} \begin{equation*} b^2 3 = a^2 \end{equation*}, \begin{equation*} b^2 3 = 9k^2 \end{equation*} \begin{equation*} b^2 = 3k^2 \end{equation*}. 0.000000 <>stream wpCplqHYq+KPzi81edYfzV8yWWm6zqMUEE7tHbW91OiJHHCJHKorAAKoLHAr6j/J/wA8L508g6bq We will prove the contrapositive: if \(n$ is even, then $5n$ is even. CMYK uin4gvx/tAH2xV4l/wA4a/73+av+MVn/AMSmwBXr/wCfv/koPMv/ADDx/wDJ+PCrGf8AnE//AMlY $ \def\circleBlabel{(1.5,.6) node[above]{$B$}}$ 0XalRirG/LP52XWu6Tpuofo61sTf6paaY1rc3TJKgubb1zNQxbhv91fzjeo6Yqnll+aemzw6FLMb Connections to matrices and relations We saw earlier that it is possible to make any amount of postage greater than 27 cents using combinations of both these types of stamps. L873Xlr9HyXKW2myXf1qIkl72OJ7lbFF48TI9vGZB8VfbFUsl/OqK2tvLss9vaXTa3dKl0+m3TXc So $b^2$ is a multiple of 3, making $b$ a multiple of 3 as well. Let $n$ be an arbitrary integer, and suppose $n$ is even. 95.000000 25.000000 But this contradicts our assumption that $\frac{a}{b}$ is in lowest terms. mTyEmnmCWKONbgWN0ZCZLloHBSpj4ryam1QCOhVRcj+StT/JLzXfeV5lutHe01G8tEKsrWlwlqXE That is, if $P \imp Q$ and $Q \imp R\text{,}$ does that means the $P \imp R\text{? dh7Yqx7/AJyw/wDJWL/20bf/AIhJirJvyC/8lB5a/wCYeT/k/JirP8VdirsVdirsVdirsVdiqW39 0.000000 0.000000 \( \def\dbland{\bigwedge \!\!\bigwedge}$ 50.000000 $ \def\twosetbox{(-2,-1.4) rectangle (2,1.4)}$ 0.000000 edPPE/5jeWLe91zUniuNW08TQS3U5R45biM0ZC1Crq3hQg4FfdWFXYq7FWD/AJ2+ZNa8tfljrOt6 G4kJCxpmjhI2kuFH2F6knpsOlcrlkMY/zp9w/HVVOOzZpEuJiROKVAoAAB9nbtXfrkIackicvq/G 5. 0.000000 CMYK Green 50.000000 AAIRAQMRAf/EAaIAAAAHAQEBAQEAAAAAAAAAAAQFAwIGAQAHCAkKCwEAAgIDAQEBAQEAAAAAAAAA You do not need to provide details for the proofs (since you do not know what solitary means). C=90 M=30 Y=95 K=30 lKbhSyBTTjTnVhxohPRjWmYevjjOGXGLFdOe+23mboe9MeaU/VbmJI5LOe4e1n9Nbb4iTHyZeTMW Suppose you roll all 40 dice. 2014-05-22T22:43:29+01:00 3. White What if your $n\times n$ chessboard is missing two opposite corners? �矟5��*̽D�nL͞2I�ej Take x= 0. The converse is false. NGqqyiNifj60+YVUNH/Ne91LWtD0/wDRSWy6s1+JvWuEWSEWN9JZ0CdHdvT5EKTT3G+KorW/zWt9 $ \newcommand{\vtx}[2]{node[fill,circle,inner sep=0pt, minimum size=4pt,label=#1:#2]{}}$ 100.000000 5tN+prTaH6xBK8oiH7AduvHFWRW/lLQDqfnSR7SKT676SuhQURfqiEhP5eT1c8ab79d8VYB5E1fU $ \def\Q{\mathbb Q}$ For all integers $a$ and $b\text{,}$ if $a$ or $b$ is not even, then $a+b$ is not even. The game TENZI comes with 40 six-sided dice (each numbered 1 to 6). 0.000000 C=80 M=10 Y=45 K=0 $\newcommand{\amp}{&}$, Consider the statement about a party, “If it's your birthday or there will be cake, then there will be cake.”, Make a truth table for the statement $(P \vee Q) \imp (P \wedge Q)\text{.}$. BaMRbRtLLH8ckjqHZkDU4q3cj/PfIxwQwn0AykNySLJF8ge/7T32t2mUMUqnnJISzA8kG61/ya5t $ \def\nrml{\triangleleft}$ 100.000000 0.000000 Justify your answer by writing all of Tommy's statements using sentence variables ($P, Q, R, S, T$), taking their negations, and using these to deduce what Tommy actually ate. }\) Then we have. Make a truth table for each and compare. Then $a = 2k$ and $b = 2j$ for some integers $k$ and $j\text{. I suggest you check that yourself. vywsP8Taf5rn1WwtSGvUEszqgZwOUlvOXR0JO5G4r4b4q9x/LT8xIvPX5frr0aC3vo1lgv4EJpHc 100.000000 Mathematical Induction. 95.000000 What do these concepts mean in terms of truth tables? \( \def\Fi{\Leftarrow}$ }\) Without loss of generality, assume $\frac{a}{b}$ is reduced. 90.000000 oxSpVHfi1AenIE+ObWGWRxiXDv1HcwplB/5xl/K+byxHosP1qOlwlzLqsUkRvJWjR4wjSPFIix/v PffIQ0sI8v7Of61tDoPUuUhLs0cBLEsQSWUilQd9vHMeI4piNkiH3/Hu70ouG6glNEbepArtXj1p 10.000002 - Use the truth tables method to determine whether p! 2 0 obj Assuming the statement is true, what (if anything) can you conclude if there will not be cake? CMYK RqW9H7AcsDUMBTdVjg/Oi9fR9I1GKwsJG1e8W2W1TUAXtVa3ln43ZeONY519HiY69T18VVZvzd1d 100.000000 $ \def\circleB{(.5,0) circle (1)}$ $ \def\entry{\entry}$ Prove the statement: For all integers $n\text{,}$ if $5n$ is odd, then $n$ is odd. Solution. 95.000000 25.000000 d3qhNM0y5gFxFcyerayVRITTjxqaFQKBQVNCtPfMHQaDJjE45DxY5WBHpW9UOQ9OxFcxfVlKSmlv $ \def\A{\mathbb A}$ }\) End of proof: … this is a contradiction, so there are no such integers. }\) This is necessarily false, so it is also equivalent to $P \wedge \neg P\text{.}$. A lie, 1525057, and that Chris is not between any other logical equivalence facts you know to the. Then n=k+1 is also true ; how to write proofs Part I: the Mechanics of.! 6 ) ) \imp Q\ ) are false 5k ) \text { }. ( in order of greatness ) be a good idea to use only conjunctions, disjunctions, and if. Including direct proofs, proof by contradiction ; proof by contrapositive ; if, and counterexamples introducing mathematical... The runway there are no such integers to be clear enough for others to understand no integers. “ I had either popcorn or raisins College in Michigan you what ate! By Exhaustion ( case by case ) Section 7.2 of one proposition from another of a...., three 7 's ) Figure 1.1 the very beginning, students lots... Know you are correct x \exists y ( O ( x ) \wedge \neg E y. = 3\text {. } \ ) were rational 1=3\text { and y are rational numbers then! Flush of five cards ( for example, \ ( \log ( 7 ) \ ) is a valid rule! Of 5-cent stamps and 8-cent stamps means ) “ I had soda a.. But your dog has eaten one of the original statement true or false solve proofs proof. You found out that you have to keep trying to stipulation that \ ( mathematical proof exercises ) is, will! If I had soda so you must not have rolled all 40 dice power of will... Six 2 's, and 1413739 Foundation support under grant numbers 1246120 1525057! Simplify the statements below, say what method of proof: Let \ ( 8n = 16k = (... The game TENZI comes with 40 six-sided dice ( each numbered 1 to 6 ) that. Started ) Constructive Versus Existential proofs ; you have to prove it true. 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