I have given this equation
Min(a-Q,b-W,c-E,d-R) = a-Q
Now i have to find the answer of the following equation
Min(a-K* Q, b-K* W, c-K* E, d-K* R) and k>=0
From my intuition i thinks it's always a-K*Q , But how to prove this ?
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Your intuition is correct only for $k\geq1$. It fails when $k$ lies between $0$ and $1$.
For e.g. $a=3$, $b=1$, $Q=4$, $W=1$, $k=0.5$
Here $a-Q<b-W$ but $a-kQ>b-kW$.
Following is a proof for $k\geq1$,
=> $a-b<k(Q-W)$ for $k\geq1$
Similarly for c-kE and d-kR