Lec 2 | MIT 6.042J Mathematics for Computer Science, Fall 2010
Why Mathematics Stack Exchange recommends it
Mathematics Stack Exchange askers cite it when they get stuck on a specific step in the lecture's proofs, such as the parity argument or the courtyard tiling induction.
Recommended by 0 people across 0 different months, 2014 to 2022.
What it is
Lecture 2 of MIT 6.042J (Mathematics for Computer Science, Fall 2010), taught by Tom Leighton and posted by MIT OpenCourseWare. It introduces proof by induction and works through examples including the irrationality of the square root of 2 and tiling a 2^n by 2^n courtyard with L-shaped tiles.
What people said
Cites user139175 · 2014-03 · Mathematics Stack Exchange · on “Very stupid algebra question... why is 'a' even?”
I am watching the MIT course "Mathematics for Computer Science" on YouTube right here [this video] . At about 7:20, We have the following equation: $ 2b^2 = a^2 $ He then asks, "What do we know about $ a^2 $?" He then says, "If $ a^2 $ is even then $ a $ must be even. However, any squared number is even. Why did he say t his?
Cites MayEls · 2020-03 · Mathematics Stack Exchange · on “In the following example in this MIT lecture, how did we establish that a $2^{n+1} \cdot 2^{n+1} = 4(2^n \cdot 2^n)$?”
In the following example at the minute 1:06:43 in this video: [this video] While using proof by induction, to make use of the assumption, while validating the implication/inductive step, the 2^n+1 square is divided into 4 squares each is 2^n * 2^n. Now how is that established, how is (2^n+1 * 2^n+1) = 4(2^n * 2^n)?
Cites atomic-coder · 2022-05 · Mathematics Stack Exchange · on “MIT 6.042J, Lec 2: Induction”
Question: Prove that a 2^n x 2^n courtyard can be tiled using L-shaped tiles of size 3 with a statue in the middle. (Use the Induction axiom) Click here to view the original question [this video] What I want to know is, can't we just nest a square of size 2^n x 2^n on which the inductive hypothesis has been applied in the centre of a square of size 2^(n+1) x 2^(n+1)? In that case, p(n) => p(n+1), right? Meaning every subsequent block up to n+1 will have a statue in the centre. I'm sorry if the way I phrased the questions isn't right.