https://www.khanacademy.org/math/strengthened-shs-general-math/x24dc5a0902f75de5:fourth-quarter/x24dc5a0902f75de5:propositions/v/conditional-statements-and-logical-equivalence 2024-12-03T23:53:40.067739643Z Conditional statements and logical equivalence We assert that a conditional statement is true. Let's identify the converse, inverse, and contrapositive and determine whether each of them must be true based on the original statement. Sal Khan video https://cdn.kastatic.org/googleusercontent/ZCdwTudJg6e6n-P2gsaUborP4izvMsGo71pvEVlX9dNYWcLXcP7VHkWpn2grt4TUP1KoJLQP9NswyHBuBLSFTBw Conditional statements and logical equivalence We assert that a conditional statement is true. Let's identify the converse, inverse, and contrapositive and determine whether each of them must be true based on the original statement. https://cdn.kastatic.org/ka-youtube-converted/5kojy1CHuyI.mp4/5kojy1CHuyI.mp4 344 Propositions https://www.khanacademy.org/math/strengthened-shs-general-math/x24dc5a0902f75de5:fourth-quarter/x24dc5a0902f75de5:propositions/v/biconditional-statements 2024-12-03T23:53:40.067739643Z Biconditional statements For most conditional statements, swapping the "if" and the "then" parts creates a new statement, and the new statement might not have the same truth value as the original. Biconditional statements are conditional statements that are true regardless of which clause is the "if" and which is the "then" part. Sal Khan video https://cdn.kastatic.org/googleusercontent/ZCdwTudJg6e6n-P2gsaUborP4izvMsGo71pvEVlX9dNYWcLXcP7VHkWpn2grt4TUP1KoJLQP9NswyHBuBLSFTBw Biconditional statements For most conditional statements, swapping the "if" and the "then" parts creates a new statement, and the new statement might not have the same truth value as the original. Biconditional statements are conditional statements that are true regardless of which clause is the "if" and which is the "then" part. https://cdn.kastatic.org/ka-youtube-converted/bs5YLyGHa60.mp4/bs5YLyGHa60.mp4 221 Propositions https://www.khanacademy.org/math/strengthened-shs-general-math/x24dc5a0902f75de5:fourth-quarter/x24dc5a0902f75de5:propositions/v/conditionals-pt1 2020-06-23T07:22:40Z Language: Conditionals, Part 1 Justin invites us to think about conditional sentences ("if P then Q"). Perhaps surprisingly, the question of what these sentences mean has vexed philosophers for thousands of years. In part one, Justin motivates the question and introduces one of the oldest answers to it, the material conditional theory.  Speaker: Dr. Justin Khoo, Assistant Professor, MIT Gaurav Vazirani video https://cdn.kastatic.org/googleusercontent/ihvAtjOOFKdsnZKCStHjlOmerheiTghz4YxcZ6CalIe3deAdpEuQhF_tNKjQA_YXX5Du4Htv6vZP3D4kSGcaJLHb Language: Conditionals, Part 1 Justin invites us to think about conditional sentences ("if P then Q"). Perhaps surprisingly, the question of what these sentences mean has vexed philosophers for thousands of years. In part one, Justin motivates the question and introduces one of the oldest answers to it, the material conditional theory.  Speaker: Dr. Justin Khoo, Assistant Professor, MIT https://cdn.kastatic.org/ka-youtube-converted/4Zxp2-_pLCE.mp4/4Zxp2-_pLCE.mp4 292 Propositions https://www.khanacademy.org/math/strengthened-shs-general-math/x24dc5a0902f75de5:fourth-quarter/x24dc5a0902f75de5:propositions/v/conditionals-pt-2 2020-06-23T07:22:36Z Language: Conditionals, Part 2 Justin Khoo invites us to think about conditional sentences ("if P then Q"). Perhaps surprisingly, the question of what these sentences mean has vexed philosophers for thousands of years. In part 2 of the series on conditionals, Justin discusses some of the challenges facing the material conditional theory. Speaker: Dr. Justin Khoo, Assistant Professor, M.I.T. video https://cdn.kastatic.org/googleusercontent/VWPShO-ySGEaLVT1JJW9oXIhVTuy3GXLQQn4_E_Wz2AfNqnZn4e2bWFp7QOHEFwf_XxXKt8MM9TEnwpZbCPGXIcS Language: Conditionals, Part 2 Justin Khoo invites us to think about conditional sentences ("if P then Q"). Perhaps surprisingly, the question of what these sentences mean has vexed philosophers for thousands of years. In part 2 of the series on conditionals, Justin discusses some of the challenges facing the material conditional theory. Speaker: Dr. Justin Khoo, Assistant Professor, M.I.T. https://cdn.kastatic.org/ka-youtube-converted/Xs6E-FEls1c.mp4/Xs6E-FEls1c.mp4 491 Propositions https://www.khanacademy.org/math/strengthened-shs-general-math/x24dc5a0902f75de5:fourth-quarter/x24dc5a0902f75de5:propositions/v/conditionals-pt3 2020-06-23T07:22:36Z Language: Conditionals, Part 3 In part 3 of the series on conditionals, Justin picks up where part 2 leaves off, introducing an alternative theory of conditionals: the strict conditional theory. According to the strict theory, conditionals express necessary connections between their antecedent and consequent. Justin shows how this theory avoids the problems facing the material conditional theory. However, the strict theory turns out to face a similar problem of its own! Speaker: Dr. Justin Khoo, Assitant Professor, M.I.T. video https://cdn.kastatic.org/googleusercontent/-xNyg9uR7D7rVAXnj8N4Ixu9BhVJMNOtOg1Ti1XZmxQvTdhCwYX6uOxDplIyfoCTJnV1bDj1eixOF2BG8Y-NUae4TA Language: Conditionals, Part 3 In part 3 of the series on conditionals, Justin picks up where part 2 leaves off, introducing an alternative theory of conditionals: the strict conditional theory. According to the strict theory, conditionals express necessary connections between their antecedent and consequent. Justin shows how this theory avoids the problems facing the material conditional theory. However, the strict theory turns out to face a similar problem of its own! Speaker: Dr. Justin Khoo, Assitant Professor, M.I.T. https://cdn.kastatic.org/ka-youtube-converted/Oxt1DdfT8ME.mp4/Oxt1DdfT8ME.mp4 514 Propositions https://www.khanacademy.org/math/strengthened-shs-general-math/x24dc5a0902f75de5:fourth-quarter/x24dc5a0902f75de5:propositions/v/conditionals-pt4 2020-06-23T07:22:43Z Language: Conditionals, Part 4 In this video, Justin picks up where part 3 left off.  He introduces the Conditional Assertion Theory of conditionals, which aims to resolve the problems presented for the other theories of conditionals. In the end, Justin presents yet another problem for this radical new theory. Speaker: Dr. Justin Khoo, Assistant Professor, MIT video https://cdn.kastatic.org/googleusercontent/Qiawph01kohT3SnNmxcglVZbTiK4zdaZn6tmJkKJEO50UQ2zXSSSJPClzNbpOBnVDxUrCNZ-ElKYFgXydfmy6g4 Language: Conditionals, Part 4 In this video, Justin picks up where part 3 left off.  He introduces the Conditional Assertion Theory of conditionals, which aims to resolve the problems presented for the other theories of conditionals. In the end, Justin presents yet another problem for this radical new theory. Speaker: Dr. Justin Khoo, Assistant Professor, MIT https://cdn.kastatic.org/ka-youtube-converted/W0BG49ir32E.mp4/W0BG49ir32E.mp4 545 Propositions https://www.khanacademy.org/math/class-9-tg/x06d55bfa213a79fd:proofs-in-mathematics/x06d55bfa213a79fd:mathematical-statements/v/different-forms-of-theoretical-statements 2026-02-18T21:20:35.275198076Z Different forms of theoretical statements In this video, we dive into the world of theoretical statements—the backbone of mathematical reasoning. We begin by introducing what a mathematical statement is: a declarative sentence that is either true or false, such as “If a number is even, then it is divisible by 2.” From there, we explore how to negate a statement, flipping its truth value while preserving logical structure. Next, we unpack the converse of a statement—what happens when we reverse the hypothesis and conclusion—and discuss why the converse isn’t always true, even if the original statement is. Finally, we explore the powerful technique of proof by contradiction, where we assume the opposite of what we want to prove and show that this assumption leads to a logical inconsistency. Through intuitive examples and step-by-step reasoning, this video builds your understanding of how mathematicians construct and challenge ideas using formal logic. Courses on Khan Academy are always 100% free. Start practicing—and saving your progress—now!Unit link 00:00 Introduction to theoretical statements 00:58 Negation of a statement 02:53 Converse of a statement 05:08 Proof by contradiction Khan Academy India is a nonprofit organization with the mission of providing a free, world-class education for anyone, anywhere. We have videos and exercises that have been translated into multiple Indian languages, and 15 million people around the globe learn on Khan Academy every month. Support Us: https://india.khanacademy.org/donate Created by Khan Academy Khan Academy video https://cdn.kastatic.org/googleusercontent/ZCdwTudJg6e6n-P2gsaUborP4izvMsGo71pvEVlX9dNYWcLXcP7VHkWpn2grt4TUP1KoJLQP9NswyHBuBLSFTBw Different forms of theoretical statements In this video, we dive into the world of theoretical statements—the backbone of mathematical reasoning. We begin by introducing what a mathematical statement is: a declarative sentence that is either true or false, such as “If a number is even, then it is divisible by 2.” From there, we explore how to negate a statement, flipping its truth value while preserving logical structure. Next, we unpack the converse of a statement—what happens when we reverse the hypothesis and conclusion—and discuss why the converse isn’t always true, even if the original statement is. Finally, we explore the powerful technique of proof by contradiction, where we assume the opposite of what we want to prove and show that this assumption leads to a logical inconsistency. Through intuitive examples and step-by-step reasoning, this video builds your understanding of how mathematicians construct and challenge ideas using formal logic. Courses on Khan Academy are always 100% free. Start practicing—and saving your progress—now!Unit link 00:00 Introduction to theoretical statements 00:58 Negation of a statement 02:53 Converse of a statement 05:08 Proof by contradiction Khan Academy India is a nonprofit organization with the mission of providing a free, world-class education for anyone, anywhere. We have videos and exercises that have been translated into multiple Indian languages, and 15 million people around the globe learn on Khan Academy every month. Support Us: https://india.khanacademy.org/donate Created by Khan Academy https://cdn.kastatic.org/ka-youtube-converted/iFtUgNM_zIg.mp4/iFtUgNM_zIg.mp4 362 Propositions https://www.khanacademy.org/math/strengthened-shs-general-math/x24dc5a0902f75de5:fourth-quarter/x24dc5a0902f75de5:propositions/e/conditional-statements 2024-12-17T19:11:46.927047272Z Conditional statements For conditional statements, identify statements that are logically equivalent. Recognize the converse, inverse, and contrapositive of a conditional statement. Determine whether a conditional statement is a biconditional statement. Charlie Auen exercise