00:02
Hi, here in this question we are given that let c equals to depend on f of x where f of x is a real continued function on interval of 0 .1 and further, f direct sum of g of x is equal to maximum of f f of x comma g, g of x.
00:34
For f and g, for f and g, which belongs to, maximum of f f of x, which belongs.
00:37
To c and all x belongs to interval of 0 comma now here we need to check whether the following properties are satisfied or not further we are also given that lambda f of x is equal to lambda f of x it has a scalar property and where lambda be any real number so here first one we need to check closure under addition so here here it whether it is closure under addition or not.
01:17
Closure under addition means it satisfies the binary plus operation.
01:22
So here we have the solution as yes, this property is satisfied.
01:28
Now similarly for the second one, here for the second one we are given that addition is associative.
01:35
Associative addition.
01:40
So here we'll define three functions such that f plus g plus h which cannot also be written as f plus g plus h.
01:50
So this will not affect the solution.
01:53
And similarly, we will also check with the value of the solution.
01:57
So here for this one, we have maximum of maximum of f comma g comma h which will be equal to maximum of f comma maximum of g comma edge.
02:14
In short, in both the cases we are just taking the maximized value.
02:17
So here this one is also true.
02:21
So yes, it is satisfied.
02:23
Now similarly for the third one, addition is commuting.
02:31
Now, here we know that if we do a plus b or b plus a, it is same for real number.
02:42
So here, yes, f plus g is equal to g plus h.
02:48
Here it will be maximum of f comma g.
02:52
It will be maximum of.
02:52
It will be maximum of.
02:53
Of g comma h which are equal...