I have this question below for homework in my Discrete Mathematics class. I was wondering what the difference between questions 1 and 2 are?

Is #1 asking if there is a row where all entries are true, while #2 is asking if there is a column where all entries are true?

$$\def\true{\color{blue}{\text{T}}}\def\false{\color{red}{\text{F}}} {\textbf{Problem II.1}\quad\text{Using the truth tables for three predicates $P$, $Q$, and $R$, over}\\\text{the domain $\mathcal D = \{a,b,c\}$, where the first argument is vertical (row) and the}\\\text{second is horizontal (column).}}\\ \begin{array}{c|ccc}P & a & b & c \\ \hline a & \true & \false & \true \\ b & \false & \false & \false \\ c & \true & \true & \true\end{array}\qquad\begin{array}{c|ccc}Q & a & b & c \\ \hline a & \false & \true & \false \\ b & \true & \false & \true \\ c & \true & \true & \true\end{array}\qquad \begin{array}{c|ccc} R & a & b & c \\ \hline a & \false & \false & \false \\ b & \false & \false & \false \\ c & \false & \false & \false\end{array}\\{\text{Evaluate the truth values for the statements from predicate logic below.}\\ \quad1.\quad \exists x\forall y,~ P(y,x)\\\quad2.\quad \exists x\forall y,~P(x,y)\\\quad3.\quad\exists x\exists y,~Q(x,y)\\\quad 4.\quad\exists x\forall y,~\neg R(x,y)}$$

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$P(y, x)$ is looking at the rows $y$ and the columns $x$. So question (1) is asking if there exists a column in the $P$ table such that all of the entries going down that column (i.e. all the rows) are true.

Question (2) asks the opposite since we are looking at $P(x, y)$: whether there exists a row such that all the entries are true.


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