# Supremum and infimum

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#1
How do I find infimum and suprrmum of set
S : { cosx + sinx | x - real number } without graph?
0
2 years ago
#2
(Original post by Infinity ∞)
How do I find infimum and suprrmum of set
S : { cosx + sinx | x - real number } without graph?
The infimum and supremum are basically the maximum/upperbound of the set. The least and max upperbounds.

For the function
cos(x)+sin(x)
it is relatively simple to find the maximum value, by a simple trig transformation into a single trig term like
r*sin(x+a)
Can you manage to do this?
0
2 years ago
#3
Think about the least and greatest value of sinx and cosx. So what's the glb and lub?
0
#4
(Original post by mqb2766)
The infimum and supremum are basically the maximum/upperbound of the set. The least and max upperbounds.

For the function
cos(x)+sin(x)
it is relatively simple to find the maximum value, by a simple trig transformation into a single trig term like
r*sin(x+a)
Can you manage to do this?
I think the single term for this function will be √2sin(x + a).
Since. -1≤sin(x+a)≤1
This implies -√2≤√2sin(x+a)≤√2
Are -√2 and √2 the infimum and supremum of this set? I think no.
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#5
(Original post by NotNotBatman)
Think about the least and greatest value of sinx and cosx. So what's the glb and lub?
Least value for sinx and cosx is -1 and greatest is 1.
What if i add lub and glb of both?
Then -2 will be inf and 2 will be sup. But this is not right approach. This will be true for set { sinx + cosy| x,y - real no}.
1
2 years ago
#6
(Original post by Infinity ∞)
I think the single term for this function will be √2sin(x + a).
Since. -1≤sin(x+a)≤1
This implies -√2≤√2sin(x+a)≤√2
Are -√2 and √2 the infimum and supremum of this set? I think no.
Why do you think no? Sounds ok to me.
0
2 years ago
#7
(Original post by Infinity ∞)
Least value for sinx and cosx is -1 and greatest is 1.
What if i add lub and glb of both?
Then -2 will be inf and 2 will be sup. But this is not right approach. This will be true for set { sinx + cosy| x,y - real no}.
This is not the approach I meant, perhaps stick to the other way of thinking (harmonic form) because root 2 is correct . Sqrt(1^2+1^2) = sqrt(2)
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#8
(Original post by mqb2766)
Why do you think no? Sounds ok to me.
Okay then supremum and infimum for this set are respectively √2 and -√2.
1
2 years ago
#9
(Original post by Infinity ∞)
Okay then supremum and infimum for this set are respectively √2 and -√2.
Yes.
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#10
(Original post by NotNotBatman)
This is not the approach I meant, perhaps stick to the other way of thinking (harmonic form) because root 2 is correct . Sqrt(1^2+1^2) = sqrt(2)
-√2 and √2 are glb and lub. Right?
0
2 years ago
#11
(Original post by Infinity ∞)
-√2 and √2 are glb and lub. Right?
Yes. But perhaps you're required to prove this (otherwise it's a pointless question).
1
11 months ago
#12
Suprimum-√2 and imf-(-√2)
0
11 months ago
#13
(Original post by Subhajit1134)
Suprimum-√2 and imf-(-√2)
You resurrect a year old thread to repeat what's already in the thread, and don't even spell it right. ffs.
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