The meaning of implication

Question

We have some predefined functions and those are

$M(x) = x$ is a woman

$D(x,y) = x$ is child of $y$

$P(x,y) = x$ is husband/wife of $y.$

And the task is to ofc in predicat. logic write these relation ships

$x$ is daughter of $y$

$x$ is grandfather of $y$

$x$ is aunt of $y$

The question is , what is right choice of logical operator in this$?$ Its an implication or conjunction. $=$

$x$ is daughter of $y$>>$D(x,y) \Rightarrow M(x)$

$x$ is grandfather of $y$>> $D(z,x) \wedge D(y,z) \Rightarrow M(y) \wedge M(z)'$

$x$ is aunt of $y$ >> $D(x,z) \wedge D(u,z) \Rightarrow D(y,u) \wedge M(x) $

does implication fits in those expression , if not , why$?$

Answer 1

Intuitively,

• conjunction represents "and"
• implication represents "if ... then ..."

For example, you should have $D(x,y) \wedge M(x)$ for the first question, as otherwise you are saying "IF $x$ is a child of $y$, THEN $x$ is a woman." What you really want to say is "$x$ is a child of $y$ AND $x$ is a woman."

You have mistakes in your other two answers as well; see if you can fix them.