Discrete Math - 1.8.1 Proof by Cases

Exploring a method of proof by exhaustion known as proof by cases.

Video Chapters:
Introduction 0:00
What is a Proof by Cases? 0:10
Proof by Cases Example 1 2:27
Proof by Cases Example 2 (Implication) 4:52
Proof by Cases Example 3 (Challenging) 9:25
Up Next 18:34

Textbook: Rosen, Discrete Mathematics and Its Applications, 7e

[KevinChau-kv]

Just to clarify for anyone about to take the test and to not make a mistake like I did, zero is not considered as even or odd is wrong. Zero is considered as an even number.

[abdurrehmansiddiqui]

Your lectures helped me greatly with my discrete maths preps. Thanks for existing ❤️

[Ravi]

Thank you so much! I am self studying proofs and linear algebra, and these videos are incredible. The quality of instruction here is incredible!

[oximas]

5:40 zero is even, an even number is defined as any number divisible by 2 aka it can be formed as 2k for k an integer

which is the case for zero

[pradmandal]

Hi! Just to confirm, in the end, you say it is a proof by contradiction and contraposition. Did you mean contraposition and cases?

[zulumopuku]

Like always, great content mom. Do you suggest any good book on the subject ?

[RenaudAlly]

Just putting a personal note here for future reference. The last example is a proof by contraposition and cases. So we can show the overall statement is true if the conclusion in that implication is true given we take into account the implication that not r is true i.e. both x and y are not even

Also note that "both x and y are not even" could mean that "both are odd" and also "either x or y are odd". That's why we use the proof by cases to take all into account. Before using Without Loss of Generality to conclude that we can assume x to be odd. Again, we can do this because it doesn't matter in this context if "x is odd" or "y is odd", because the result will be the same.

[avrillellaine]

Hello ma'am! May I ask if proof by cases is the same as the choose method? I’m confused about the two. The same goes for the Construction method and direct proofs. Are those two the same as well? Please enlighten me. Thanks.

[wintutorials]

3:55 why can you multiply both sides with n as a proof for case 3 but not for case 1? Case 1 would give n^2<-n ??

I think it's better to say that

for case 1, if n <= -1 and you subsititute any arbitrary integer that satisfies n <= -1, n^2 > 0 and we know n <= -1, so n <=n^2

(for case 2, direct substitution is possible so that speaks for itself)

for case 3, if n>=1 i don't know...

how can i prove??? i confused

[sankalppatil]

At the end when you said this example made use of contraposition and proof by contradiction did you mean contrapositions and proof by cases? I don't see and proof by contradiction in that problem.

[abrilthom]

hi, doing it by the direct method, how did the demonstration had been

[victoriawinston]

Sorry, could you clarify why you didn't need to do case 4 for the last question? (If x and y are integers and both xy and x+y are even, then both x and y are even")

[ramielezzy]

Hi!! Thanks a lot for this video it was very helpful, but i got a question, we negated R which is 'both X and Y are even', so the negation of this means 'both X and Y are odd' so why did we have multiple cases instead of just the case where both x and y are odd?

[kevingerges]

is there is a proof by induction video?

[cowbell]

how do we know that 2r and 2L is even based on the algebraic equation? 9:10