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There is a really intuitive way to understand the "X is nessary for Y". "X is nessary for Y" is equal to "If you want Y, you must have X". So it can be written as q --> p.
Here is one way in which one might make translation of the sentences with "necessary" sort of more intuitive. First consider the sentence "if it is not sunny then John is not outside". Observe that it says pretty much the same thing as "If John is outside, then it is sunny". Here it looks like there is an equivalence between "not p -> not q" and "q -> p". Consider a natural language sentence of the form "A is necessary for B". This seems to be equivalent to saying that "If A is not the case, then B is not the case". Assuming that A translates into p and B translates into q, the natural language sentence seems to have the following: translation "not p -> not q". This, we have just seen, is equivalent to "q -> p". That is, the natural language sentence "A is necessary for B" (where A translates into p and B translates into q) translates into "q -> p".
I watch a playlist on a topic until I cannot understand, then I start another playlist from scratch on the same topic. I have learned a lot using this method. But I must say that you have cleared up so much for me.
The statement: "17y + 20x is an integer" shown at 4:10 is a statement. It's what's called an "open statement" in discrete mathematics. So if you answered true, you are not wrong. It's just that we can't tell what the outcome of the statement will be. Don't believe me? Just do a quick search for "Open statements in discrete mathematics".
Sorry, I'm still a little confused about #2 @2:01. If q can't happen without p, why isn't it p->q. Can't that sentence also mean if I finish writing my program before lunch, then I can play tennis this afternoon?
Hm, not sure, but when you're doing if/then statement, you're not saying that q can't happen without p, just that q may not happen without p, but q can indeed still happen! So, just because you don't finish writing your program before lunch, doesn't necessarily mean you're not gonna go play some tennis :) Consider, q->p, where the only case in which you will ever be able to play tennis is if you finished writing your program. In this case, where p is necessary, it is also possible that you may not play tennis, :( What you're thinking of probably is "if and only if" where it is necessary for the conclusion (q) and the hypothesis (p) to both have the same truth value.
Implications are confusing man, that's ok to feel so. The arrow of implication may go in the opposite direction to physical causality. For example Raining -> Cloudy is a good relation and Cloudy -> Raining is not. Why? See: Raining -> Cloudy is the same as ~(Raining & ~Cloudy) which means "It can't be raining with no clouds" which is correct. Cloudy -> Raining is the same as ~(Cloudy & ~Raining) which means "It can't be cloudy without raining" which is bogus. Another useful formula: a -> b ~b -> ~a. Let a ="Finished program", b = "Playing tennis". Now "If not playing tennis then not finished program" - bullshit. "If not finished program then not playing tennis" - legit. So ~a -> ~b, hence b -> a.
No it cant, if you finish you program(base condition) then you can or can not play tennis (optional). However, if you are playing tennis, then you must have finished your program(since it is necessary).
i understand how u explain nub 2 but then on 3 i can also write the opposit q -> (s AND r) cz if i am playing tennis it is necessary that low humidity and sunshine . but if it is low humidity and sunshine then it is not necessary that i am playing tennis. ??
The second example at the end would depend on how the sentence is defined I would think. If you were asking if 17y+20x was a integer as opposed to a expression, then it could be a statement. However if you were saying if the result of the expression 17y+20x is a integer, then it would be a function of x and y and it would in fact not be a statement.
1:45 can I say: `if not p then not q`? is it equivalent to `if q then p`? p = Finishing the writing of my computer program before lunch q = my playing tennis this afternoon
i got wrong 2 and 3. After seeing the solutions my conclusion is that when i see the keyword 'necessary' the answer will always be q->p (second part of the sentence(q) will come first then the arrow -> then p). When i see the keyword 'sufficient' i just write down symbols left to right s, r, ->, q. If there is a 'not' keyword in the sentence i suppose it doesn't change the order that p, q, -> will appear on a paper.
I am still confused about how answer 2 is q -> p. I read the question as if I finish the writing of my computer program before lunch, then I will play tennis this afternoon. The current solution doesn't make intuitive sense to me. Please, what am I missing?
If you do not finish writing your computer program (0) and play tennis (1), then the statement is true under your interpretation. But I said that writing the computer program is necessary for playing tennis, which means it should be false, not true.
TheTrevTutor by the truth table of p-q he seems right and your wrong. I tried to do with q-r but if q is t and r is f then the value will be false opposite to you
Just confused of the 4th statement problem. I think that "I can't live without you" is an opinion and I guess opinions cant be treated as true or false. I could be wrong so just asking for the clarification. Thank you.
"If john manages his time well and studies hard then he will obtain good grades and have a better future." How would you write the negation, Inverse, converse and contrapositive of this compound statement?
A= John manages his time well; B= John studies hard; C=John obtains good grades; D= John has a better future (A *and* B) -> (C *and* D) I don't know how to type the sign for *and* so I put it in asterisks. Hope this helps :)
Assume some statement in p -> q form. "sufficient" means statement is wrong when p=1 and q=0 (wasn't sufficient). "necessary" means statement is wrong when p=0 and q=1 (wasn't necessary). Implications are easier to get from the opposite because of their truth table.
I think your solution for sentence 2 isn't really a representation of the sentence. q -> p would translate into "Under the condition of finishing the writing of my computer program before lunch I will play tennis this afternoon" But the actual sentence doesn't say that you are really going to play tennis. It just says "q would be possible if p". But if it is going to happen can't be concluded just by the sentence. Edit: Same thing for 3. These only state conditions that need to be true in order for it to be possible. There is still a condition left, it's not clear if it is going to happen.
Hello... Can you help me understand why p is sufficient for q thus p -> q for question 3 when p: I finish writing my computer program before lunch but not Low humidity and sunshine
If we rearrange the 2nd example, we can get, "If I finish writing of my computer program before lunch, then I will play tennis this afternoon". Then the translation will be p -> q. Why will it be wrong?
How is “Tell me the time” not a statement? But “I want you to tel me the time” is a statement? They’re both commands. They both can be accepted or denied.
For #2, since if finishing the writing of my computer program was false(q), wouldn't that make playing tennis this afternoon(p) true according to a truth table of q implies p?
+Submersed24 Well, we aren't computing truth values for that one, but the p's and q's you have written are switched around. If the antecedent is false, the conditional is true.
The following two statements sounds equivalences, aren't they!? “If you do your homework, you will not be punished” “If you do not do your homework, you will be punished.”
Natural language tends to conflate two weaker logical if-then into a stronger if-and-only-if-then. If both your statements are applicable, it means you're actually saying "If and only if you do you home work you won't be punished".
2nd example does not make sense. if you play tennis then you will finish the computer program before lunch? WTF.. even if you put it in reverse. it is necessary to finish my com program before lunch so i play tennis. but when you make first p false and q true according to the truth table. then it says that it is NOT necessary for you to finish the program to play tennis. XD
Those are two facts that are both true right? What happens if one is false in the compound? What happens if both are false? What happens if both are true? Sounds like a conjunction.
![[Discrete Mathematics] Negating Quantifiers and Translation Examples](http://i.ytimg.com/vi/7HvCgm4vBv4/mqdefault.jpg)
Check out my new course in Propositional Logic: trevtutor.com/p/master-discrete-mathematics-propositional-logic
It comes with video lectures, text lectures, practice problems, solutions, and a practice final exam!
Khan Academy should hire you to helm a Discrete Mathematics section for their site. Awesome vids man. Thank you.
Definitely
Much needed at Khan Academy, they lack content ahead of high-school lvl
There is a really intuitive way to understand the "X is nessary for Y". "X is nessary for Y" is equal to "If you want Y, you must have X". So it can be written as q --> p.
Here is one way in which one might make translation of the sentences with "necessary" sort of more intuitive. First consider the sentence "if it is not sunny then John is not outside". Observe that it says pretty much the same thing as "If John is outside, then it is sunny". Here it looks like there is an equivalence between "not p -> not q" and "q -> p". Consider a natural language sentence of the form "A is necessary for B". This seems to be equivalent to saying that "If A is not the case, then B is not the case". Assuming that A translates into p and B translates into q, the natural language sentence seems to have the following: translation "not p -> not q". This, we have just seen, is equivalent to "q -> p". That is, the natural language sentence "A is necessary for B" (where A translates into p and B translates into q) translates into "q -> p".
thank you most logical explanation for this case
You blew my mind with this explanation. Thank you so much it's so simple now.
But we can also say, "If it is sunny, then John is outside." Then it will be p -> q. Why will it be wrong? It is saying the same thing.
Best teacher for Discrete Mathematics. Thank you.
I watch a playlist on a topic until I cannot understand, then I start another playlist from scratch on the same topic. I have learned a lot using this method. But I must say that you have cleared up so much for me.
The statement: "17y + 20x is an integer" shown at 4:10 is a statement.
It's what's called an "open statement" in discrete mathematics. So if you answered true, you are not wrong.
It's just that we can't tell what the outcome of the statement will be.
Don't believe me? Just do a quick search for "Open statements in discrete mathematics".
I almost choked on my coffee when he said it’s not a statement
@@SamraiCast mmmh coffee
Sorry, I'm still a little confused about #2 @2:01. If q can't happen without p, why isn't it p->q. Can't that sentence also mean if I finish writing my program before lunch, then I can play tennis this afternoon?
exactly
Hm, not sure, but when you're doing if/then statement, you're not saying that q can't happen without p, just that q may not happen without p, but q can indeed still happen! So, just because you don't finish writing your program before lunch, doesn't necessarily mean you're not gonna go play some tennis :)
Consider, q->p, where the only case in which you will ever be able to play tennis is if you finished writing your program. In this case, where p is necessary, it is also possible that you may not play tennis, :(
What you're thinking of probably is "if and only if" where it is necessary for the conclusion (q) and the hypothesis (p) to both have the same truth value.
Implications are confusing man, that's ok to feel so. The arrow of implication may go in the opposite direction to physical causality. For example Raining -> Cloudy is a good relation and Cloudy -> Raining is not. Why? See: Raining -> Cloudy is the same as ~(Raining & ~Cloudy) which means "It can't be raining with no clouds" which is correct. Cloudy -> Raining is the same as ~(Cloudy & ~Raining) which means "It can't be cloudy without raining" which is bogus. Another useful formula: a -> b ~b -> ~a. Let a ="Finished program", b = "Playing tennis". Now "If not playing tennis then not finished program" - bullshit. "If not finished program then not playing tennis" - legit. So ~a -> ~b, hence b -> a.
No it cant, if you finish you program(base condition) then you can or can not play tennis (optional). However, if you are playing tennis, then you must have finished your program(since it is necessary).
i understand how u explain nub 2 but then on 3 i can also write the opposit q -> (s AND r) cz if i am playing tennis it is necessary that low humidity and sunshine . but if it is low humidity and sunshine then it is not necessary that i am playing tennis. ??
Just waiting for Obama to drop that mixtape
OBAMA GANGNAM STYLE
The second example at the end would depend on how the sentence is defined I would think. If you were asking if 17y+20x was a integer as opposed to a expression, then it could be a statement. However if you were saying if the result of the expression 17y+20x is a integer, then it would be a function of x and y and it would in fact not be a statement.
Hi, can anyone explain to me why statements do not have variables? I'm sorry if the question sounds stupid but thank you to the one who will answer!
1:45 can I say: `if not p then not q`? is it equivalent to `if q then p`?
p = Finishing the writing of my computer program before lunch
q = my playing tennis this afternoon
i got wrong 2 and 3. After seeing the solutions my conclusion is that when i see the keyword 'necessary' the answer will always be q->p (second part of the sentence(q) will come first then the arrow -> then p). When i see the keyword 'sufficient' i just write down symbols left to right s, r, ->, q. If there is a 'not' keyword in the sentence i suppose it doesn't change the order that p, q, -> will appear on a paper.
You are a great teacher !
Wow
Amazing explanations
5:20 shouldn't be the 4) example "I can live without you"? because of the fact that we cannot write a statement with NOT?
"Tell me the time."
"No."
XDDDD Sorry I can't help it I just found that funny for some reason....
What do you do if there is while tho? Eg:
A: bob is happy
B: dora is high
"Dora is high while bob is happy" what to do?
I am still confused about how answer 2 is q -> p. I read the question as if I finish the writing of my computer program before lunch, then I will play tennis this afternoon. The current solution doesn't make intuitive sense to me. Please, what am I missing?
If you do not finish writing your computer program (0) and play tennis (1), then the statement is true under your interpretation. But I said that writing the computer program is necessary for playing tennis, which means it should be false, not true.
I see what you mean. Thanks
why it should be false? I'm still confused...
Yeah why it should be false?
TheTrevTutor by the truth table of p-q he seems right and your wrong. I tried to do with q-r but if q is t and r is f then the value will be false opposite to you
Just confused of the 4th statement problem. I think that "I can't live without you" is an opinion and I guess opinions cant be treated as true or false. I could be wrong so just asking for the clarification. Thank you.
I think I can elucidate this for you. That opinion is stated, therefore it is a statement.
"I cant live without you" is a compound of two statements. I cant live without you would be -q --> -p.
Correct me if im wrong.
It depends on what p and q are
wait, isn't the second sentence qp??
What if we had ,"is necessary and sufficient in one line" would it be implication or bi-implication.
"If john manages his time well and studies hard then he will obtain good grades and have a better future."
How would you write the negation, Inverse, converse and contrapositive of this compound statement?
A= John manages his time well; B= John studies hard; C=John obtains good grades; D= John has a better future
(A *and* B) -> (C *and* D)
I don't know how to type the sign for *and* so I put it in asterisks. Hope this helps :)
Keep teaching us please
what if its Low humidity and sunshine are necessary for my playing tennis this afternoon? Thank you ~
whats the difference between "necessary" and "sufficient"?
Assume some statement in p -> q form. "sufficient" means statement is wrong when p=1 and q=0 (wasn't sufficient). "necessary" means statement is wrong when p=0 and q=1 (wasn't necessary). Implications are easier to get from the opposite because of their truth table.
I think your solution for sentence 2 isn't really a representation of the sentence.
q -> p would translate into "Under the condition of finishing the writing of my computer program before lunch I will play tennis this afternoon"
But the actual sentence doesn't say that you are really going to play tennis. It just says "q would be possible if p". But if it is going to happen can't be concluded just by the sentence.
Edit:
Same thing for 3. These only state conditions that need to be true in order for it to be possible. There is still a condition left, it's not clear if it is going to happen.
Example of Question 2:
Books are necessary for studies.
Does that mean for you to study you need books even though you can study without books -_-??
Hello...
Can you help me understand why p is sufficient for q thus p -> q for question 3 when p: I finish writing my computer program before lunch but not Low humidity and sunshine
about the second sentence on 1:53, can't it be if and only if? as in the two sides arrow ↔
That would be "is necessary and sufficient for"
OK got it thank you very much and God bless you 😊
If we rearrange the 2nd example, we can get, "If I finish writing of my computer program before lunch, then I will play tennis this afternoon". Then the translation will be p -> q. Why will it be wrong?
4:11 That's a very beautiful Yes you wrote there.
How is “Tell me the time” not a statement? But “I want you to tel me the time” is a statement? They’re both commands. They both can be accepted or denied.
"Tell me the time" cannot be true or false, whereas "I want you to tell me the time" can, at least to my understanding
"I want you to tell me the time" is not a command. It's a fact. Wanting you to turn off the lights, and you turning off the lights are not related.
For #2, since if finishing the writing of my computer program was false(q), wouldn't that make playing tennis this afternoon(p) true according to a truth table of q implies p?
+Submersed24 Well, we aren't computing truth values for that one, but the p's and q's you have written are switched around. If the antecedent is false, the conditional is true.
Crazy world.
This is great.
Thank U 🙏
Number (1) is definitely a true statement.
0:00
3:27
The following two statements sounds equivalences, aren't they!?
“If you do your homework, you will not be punished”
“If you do not do your homework, you will be punished.”
No. They are inverses of each other.
but in natural language, both say the very same thing, isn't it!?
Natural language tends to conflate two weaker logical if-then into a stronger if-and-only-if-then. If both your statements are applicable, it means you're actually saying "If and only if you do you home work you won't be punished".
In the first case you may not do your homework and also not be punished while in the second case you may do your homework and still get punished lol
1:54 I really don't understand the logic...
day 1 of 3 {studying for final}
I can't live withoot you :)
Kinda wanna listen to an Obama Kpop album
Is this a statement?
In 1999, Barack Obama told me that 17y & 20x are Integers.
I would say so. That is True or False and does not depend on what the variable x and y actually are.
TheTrevTutor Why can't variables be statements, they could be integers or could could not?
2nd example does not make sense.
if you play tennis then you will finish the computer program before lunch? WTF..
even if you put it in reverse.
it is necessary to finish my com program before lunch so i play tennis.
but when you make first p false and q true according to the truth table. then it says that it is NOT necessary for you to finish the program to play tennis. XD
.
all the teachers should be fired, use tutors to instead them
What do you do if there is while tho? Eg:
A: bob is happy
B: dora is high
"Dora is high while bob is happy" what to do?
Those are two facts that are both true right? What happens if one is false in the compound? What happens if both are false? What happens if both are true?
Sounds like a conjunction.