Some folks are confused by the “math magic” step (step 3), so I will clarify: The first thing to realize is that the professor cancels the 1 with one of two -1, which is why step 3 contains the remaining -1. Now, the magic part is nothing but simple intuition. Notice that the professor denotes step 3 as ≤ left of the equation. This tells us that step 2 (the inductive hypothesis) is less than or equal to step 3. This means that no matter what h(T1) and h(T2) are, their sum will always be less than or equal to 2 * max( h(T1), h(T2) ). The sum is equal only when h(T1) = h(T2). In other words: Let n = h(T1) = h(T2), all positive integers Therefore, 1 + (2^[h(T1) + 1] - 1) + (2^[h(T2) + 1] - 1) = 2 * 2^[n + 1] - 1 Now let n = max( h(T1), h(T2) ) and h(T1) ≠ h(T2) Therefore, 1 + (2^[h(T1) + 1] - 1) + (2^[h(T2) + 1] - 1) ≤ 2 * 2^[max( h(T1), h(T2) ) + 1] - 1 I have my final tomorrow, so this helped a lot. Thank you for your clear explanations professor.
"Math magic" following a formula and then hearing to work it out you have to pull something out of thin air. That right there explains my burning-lava-of -rage dislike to Math
thank you for your effort, mrs. Brehm.
Some folks are confused by the “math magic” step (step 3), so I will clarify:
The first thing to realize is that the professor cancels the 1 with one of two -1, which is why step 3 contains the remaining -1.
Now, the magic part is nothing but simple intuition. Notice that the professor denotes step 3 as ≤ left of the equation. This tells us that step 2 (the inductive hypothesis) is less than or equal to step 3. This means that no matter what h(T1) and h(T2) are, their sum will always be less than or equal to 2 * max( h(T1), h(T2) ). The sum is equal only when h(T1) = h(T2). In other words:
Let n = h(T1) = h(T2), all positive integers
Therefore, 1 + (2^[h(T1) + 1] - 1) + (2^[h(T2) + 1] - 1) = 2 * 2^[n + 1] - 1
Now let n = max( h(T1), h(T2) ) and h(T1) ≠ h(T2)
Therefore, 1 + (2^[h(T1) + 1] - 1) + (2^[h(T2) + 1] - 1) ≤ 2 * 2^[max( h(T1), h(T2) ) + 1] - 1
I have my final tomorrow, so this helped a lot. Thank you for your clear explanations professor.
huh
real
same man...same
Thank you mrs u really helped me out im forever in your dept
"Math magic" following a formula and then hearing to work it out you have to pull something out of thin air. That right there explains my burning-lava-of -rage dislike to Math
100% agree
But miss your n and h typing looks same so it's a little confusion for me to understand
Small Typo: "A" should be the set of all *positive* integers divisible by 7