Proof by contradiction
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This document discusses proof by contradiction, a form of indirect proof used to establish the truth of a statement. It does this by assuming the statement is false and then showing this assumption leads to a logical contradiction, proving the statement must be true. The document provides examples of using proof by contradiction to show a number is irrational, and to prove statements about even and odd integers. It also outlines the basic steps of proof by contradiction.
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Proof by contradiction
This document discusses proof by contradiction, an indirect proof method. It provides examples of using proof by contradiction to prove different mathematical statements. The key steps in a proof by contradiction are: 1) assume the statement to be proved is false, 2) show that this assumption leads to a logical contradiction, and 3) conclude that the original statement must be true since the assumption was false. The document provides examples of proofs by contradiction for statements such as "there is no greatest integer" and "if n is an integer and n3 + 5 is odd, then n is even."
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The document discusses proofs by contraposition. It explains that a statement of the form "if p then q" can be proven by showing its contrapositive "if not q then not p" is true. It provides examples of proofs using this method, including proving if n^2 is even then n is even, if m + n is even then m and n have the same parity, and if 3n + 2 is odd then n is odd. Homework exercises are provided applying this proof technique.
Method of direct proof
This document discusses different types of proofs in mathematics including direct proofs, existence proofs, and disproving statements with counterexamples. It provides examples of direct proofs showing that the sum of two even integers is even and that the square of an odd integer is odd. It also gives an example of an existence proof showing there is an even number that can be written as the sum of two prime numbers. Finally, it discusses how to disprove universal statements by finding a counterexample.
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- Tautologies and contradictions are types of statements that are always true or false regardless of variable values.
- Quantifiers like "for all" and "there exists" can be used to define propositional functions on a domain.
- Valid arguments are those where the conclusion is necessarily true
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Proof by contradiction
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4- هُنالك بعض المعلومات تم توضيحها بشكل تفصيلي جداً (تُعتبر لدى الطالب أو الطالبة بإنها معلومات مُبهمة ومع ذلك تم توضيح هذهِ المعلومات المُبهمة بشكل تفصيلي جداً
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6- تحتوي الملزمة في اول سلايد على خارطة تتضمن جميع تفرُعات معلومات الجهاز الهيكلي المذكورة في هذهِ الملزمة
واخيراً هذهِ الملزمة حلالٌ عليكم وإتمنى منكم إن تدعولي بالخير والصحة والعافية فقط
كل التوفيق زملائي وزميلاتي ، زميلكم محمد الذهبي 💊💊
🔥🔥🔥🔥🔥🔥🔥🔥🔥
Information and Communication Technology in Education
(𝐓𝐋𝐄 𝟏𝟎𝟎) (𝐋𝐞𝐬𝐬𝐨𝐧 2)-𝐏𝐫𝐞𝐥𝐢𝐦𝐬
𝐄𝐱𝐩𝐥𝐚𝐢𝐧 𝐭𝐡𝐞 𝐈𝐂𝐓 𝐢𝐧 𝐞𝐝𝐮𝐜𝐚𝐭𝐢𝐨𝐧:
Students will be able to explain the role and impact of Information and Communication Technology (ICT) in education. They will understand how ICT tools, such as computers, the internet, and educational software, enhance learning and teaching processes. By exploring various ICT applications, students will recognize how these technologies facilitate access to information, improve communication, support collaboration, and enable personalized learning experiences.
𝐃𝐢𝐬𝐜𝐮𝐬𝐬 𝐭𝐡𝐞 𝐫𝐞𝐥𝐢𝐚𝐛𝐥𝐞 𝐬𝐨𝐮𝐫𝐜𝐞𝐬 𝐨𝐧 𝐭𝐡𝐞 𝐢𝐧𝐭𝐞𝐫𝐧𝐞𝐭:
-Students will be able to discuss what constitutes reliable sources on the internet. They will learn to identify key characteristics of trustworthy information, such as credibility, accuracy, and authority. By examining different types of online sources, students will develop skills to evaluate the reliability of websites and content, ensuring they can distinguish between reputable information and misinformation.
How to Manage Reception Report in Odoo 17
A business may deal with both sales and purchases occasionally. They buy things from vendors and then sell them to their customers. Such dealings can be confusing at times. Because multiple clients may inquire about the same product at the same time, after purchasing those products, customers must be assigned to them. Odoo has a tool called Reception Report that can be used to complete this assignment. By enabling this, a reception report comes automatically after confirming a receipt, from which we can assign products to orders.
مصحف القراءات العشر أعد أحرف الخلاف سمير بسيوني.pdf
مصحف أحرف الخلاف للقراء العشرةأعد أحرف الخلاف بالتلوين وصلا سمير بسيوني غفر الله له
KHUSWANT SINGH.pptx ALL YOU NEED TO KNOW ABOUT KHUSHWANT SINGH
INDIA`S OWN LITERARY GENIUS MR.KHUSHWANT SINGH WAS TRULY A VERY BRAVE SOUL AND WAS AWARDED WITH THE MAGIC OF WORDS BY GOD.
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【网址🎉ac44.net🎉】欧洲杯下注在体育博彩方面，不难发现，欧洲杯下注多为英式运动提供投注。欧洲杯下注为丰富多彩的体育项目提供投注，如足球、赛马、板球和飞镖，但这不是欧洲杯下注的唯一优势。在欧洲杯下注，全世界玩家都可以找到自己感兴趣的比赛进行投注。欧洲杯下注为网球和美式运动如棒球、美式足球、篮球、冰球开出的投注同样值得关注。在夏季或冬季奥运会方面，欧洲杯下注为玩家开出了丰富多彩的服务，是你的不二之选。你还可以投注水球、自行车和班迪球等小众体育项目，甚至可以为武术、拳击和综合格斗投注。欧洲杯下注为众多体育项目开出滚球投注。
A Free 200-Page eBook ~ Brain and Mind Exercise.pptx
(A Free eBook comprising 3 Sets of Presentation of a selection of Puzzles, Brain Teasers and Thinking Problems to exercise both the mind and the Right and Left Brain. To help keep the mind and brain fit and healthy. Good for both the young and old alike.
Answers are given for all the puzzles and problems.)
With Metta,
Bro. Oh Teik Bin 🙏🤓🤔🥰
Elevate Your Nonprofit's Online Presence_ A Guide to Effective SEO Strategies...
Whether you're new to SEO or looking to refine your existing strategies, this webinar will provide you with actionable insights and practical tips to elevate your nonprofit's online presence.
THE SACRIFICE HOW PRO-PALESTINE PROTESTS STUDENTS ARE SACRIFICING TO CHANGE T...
The recent surge in pro-Palestine student activism has prompted significant responses from universities, ranging from negotiations and divestment commitments to increased transparency about investments in companies supporting the war on Gaza. This activism has led to the cessation of student encampments but also highlighted the substantial sacrifices made by students, including academic disruptions and personal risks. The primary drivers of these protests are poor university administration, lack of transparency, and inadequate communication between officials and students. This study examines the profound emotional, psychological, and professional impacts on students engaged in pro-Palestine protests, focusing on Generation Z's (Gen-Z) activism dynamics. This paper explores the significant sacrifices made by these students and even the professors supporting the pro-Palestine movement, with a focus on recent global movements. Through an in-depth analysis of printed and electronic media, the study examines the impacts of these sacrifices on the academic and personal lives of those involved. The paper highlights examples from various universities, demonstrating student activism's long-term and short-term effects, including disciplinary actions, social backlash, and career implications. The researchers also explore the broader implications of student sacrifices. The findings reveal that these sacrifices are driven by a profound commitment to justice and human rights, and are influenced by the increasing availability of information, peer interactions, and personal convictions. The study also discusses the broader implications of this activism, comparing it to historical precedents and assessing its potential to influence policy and public opinion. The emotional and psychological toll on student activists is significant, but their sense of purpose and community support mitigates some of these challenges. However, the researchers call for acknowledging the broader Impact of these sacrifices on the future global movement of FreePalestine.
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Plenary presentation at the NTTC Inter-university Workshop, 18 June 2024, Manila Prince Hotel.
A Visual Guide to 1 Samuel | A Tale of Two Hearts
These slides walk through the story of 1 Samuel. Samuel is the last judge of Israel. The people reject God and want a king. Saul is anointed as the first king, but he is not a good king. David, the shepherd boy is anointed and Saul is envious of him. David shows honor while Saul continues to self destruct.
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Information and Communication Technology in Education
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Proof by contradiction
- 1. FERDINAND D. AGDORO JR. PROOF BY CONTRADICTION
- 2. 2+2 =?
- 3. 2+2 =45 1 2+2 = 2+2 =
- 5. PROOF BY CONTRADICTION *PROOF - proof is a rhetorical device for convincing someone else that a mathematical statement is true or valid. *CONTRADICTION - a proposition, statement, or phrase that asserts or implies both the truth and falsity of something PROOF BY CONTRADICTION -- a form of proof, and more specifically a form of indirect proof, that establishes the truth or validity of a proposition by showing that the proposition's being false.
- 6. PROOF BY CONTRADICTION This is not limited to proving just conditional statements – it can be used to prove any kind of statement whatsoever. The basic idea is to assume that the statement we want to prove is false, then show that this assumption leads to nonsense. We are then led to conclude that we were wrong to assume the statement was false, so the statement must be true.
- 8. (also known as reductio ad absurdum, Latin for "by reduction to the absurd"), it is shown that if some statement were true, a logical contradiction occurs, hence the statement must be false. PROOF BY CONTRADICTION
- 9. A real number x is RATIONAL if x = a/b for some a, b are integers. Also, x is IRRATIONAL if it is not rational, that is if x ≠ a/b for every a, b are integers. DEFINITION STEPS EXAMPLE PROVE BY CONTRADICTION is an irrational number is an irrational number PROPOSITIO N PROOF
- 10. PROOF
- 11. EXAMPLE PROVE BY CONTRADICTION Suppose a is an integer. If a2 is even, then a is even.
- 12. Suppose n is an integer. If n is odd, then n2 is odd.
- 13. QUESTIONS
- 16. THANK YOU !