In my Discrete Math class we recently started discussing Min

In my Discrete Math class we recently started discussing Min and Max and I am so confused on how to do this

Propistion: For an element of all real numbers k,l,n prove the following statement

min(max(k,l),max(k,n))=max(k,min(l,n))

Please help I don\'t understand it

Solution

Consider any 3 real numbers k,l,n

Let us assume k>l>n.

Then max(k,l) = k

min(max(k,l)) , max (k,n)= min k,k =k

Right side max {k, min(l,n)} = max {k,n} =k

Similarly let us assume now

k>l and K<n and i.e. n>k>l

Then left side

min {max(k,l), max (k,n)} = min {k,k} =k

Right side = max {k,min(l,n)} = max(k,l) = k

Hence proved for this also.

Now let n<l<k

Similar steps this also can be proved

Hence for all possibilities

min(max(k,l),max(k,n))=max(k,min(l,n))