1. more contrapositive

given: if the sum of two numbers is strictly less than 50 then at least one number is strictly less than 25.

I am supposed to do this by contrapositive

Is the contrapositive, "if two numbers are equal or great than 25, then their sum is 50 or greater" ??

if so then

llet $a,b \in \mathbb{Z}, s.t. a \ge 25;b \ge 25$

then $a+b \ge 50$

therefore a or b must be less than 25 in order for the sum to be less than 50.

that's it

3. Re: more contrapositive

More work is needed in the proof imho.

If \displaystyle \begin{align*} a \geq 25 \end{align*} then \displaystyle \begin{align*} a = 25 + c \end{align*} where \displaystyle \begin{align*} c \geq 0 \end{align*}.

If \displaystyle \begin{align*} b \geq 25 \end{align*} then \displaystyle \begin{align*} b = 25 + d \end{align*} where \displaystyle \begin{align*} d \geq 0 \end{align*}.

So \displaystyle \begin{align*} a + b = 25 + c + 25 + d = 50 + c + d \geq 50 \end{align*} as \displaystyle \begin{align*} c,d \geq 0 \end{align*}.

4. Re: more contrapositive

More work is needed in the proof imho.

If \displaystyle \begin{align*} a \geq 25 \end{align*} then \displaystyle \begin{align*} a = 25 + c \end{align*} where \displaystyle \begin{align*} c \geq 0 \end{align*}.

If \displaystyle \begin{align*} b \geq 25 \end{align*} then \displaystyle \begin{align*} b = 25 + d \end{align*} where \displaystyle \begin{align*} d \geq 0 \end{align*}.

So \displaystyle \begin{align*} a + b = 25 + c + 25 + d = 50 + c + d \geq 50 \end{align*} as \displaystyle \begin{align*} c,d \geq 0 \end{align*}.