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Math Help - more contrapositive

  1. #1
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    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.
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  2. #2
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    Re: more contrapositive

    that's it
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  3. #3
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    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 $\displaystyle \begin{align*} c,d \geq 0 \end{align*}$.
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  4. #4
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    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 $\displaystyle \begin{align*} c,d \geq 0 \end{align*}$.
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