# Discrete math Question.Suppose your friend makes the following English statement "If

Discrete math Question.
Suppose your friend makes the following English statement "If $X\oplus Y$, but $\sim X$, then we have Y." Convert it into a statement form. Then show that your friend's statement is valid. Is it true that "$X\oplus Y$, but $\sim X$" is equivalent to Y?

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Step 1
The symbol $\oplus$ represent the $X\oplus R$ operation for the terms in binary operation. Therefore, the expression with XOR can be solved using the truth tables.
Step 2
To convert the English statement to a statement, write it in mathematical form by removing if and then. Also remove all other texts in statements. This will give the statement : $X\oplus Y\sim X=Y$.
To check the validity of the statement make a truth table with each entry. The first column will be X then Y followed by the operations.
$\begin{array}{|ccccccc|}\hline X& Y& X\oplus Y& \sim X& \sim X\oplus Y& X\oplus Y\cdot \sim X& X\oplus YOR\sim X\\ 0& 0& 0& 1& 1& 0& 1\\ 0& 1& 1& 1& 0& 1& 1\\ 1& 0& 1& 0& 0& 0& 1\\ 1& 1& 0& 0& 1& 0& 0\\ \hline\end{array}$
Step 3
The value of $\sim X\oplus Y$ is not equivalent to Y. Also the value of $X\oplus Y\cdot \sim X,X\oplus YOR\sim X$ are not equal to Y. Therefore, the statement is false.