Hi I am struggling with these questions any help would be appreciated. I am struggling and am not sure where to start.
Let $X = \{ 1, 2, 3 \}$ and let a relation $p$ be defined as $x\,p\,y \leftrightarrow x \leq y$
List all the pairs of the relation.
Is $p$ reflexive? Prove the answer.
Is $p$ transitive? Prove the answer.
Is $p$ symmetric? Prove the answer.
On
The list of pairs that can be related are all pairs of elements from $X$. They are $1p1$, $1p2$, $1p3$, $2p1$, $2p2$, $2p3$, $3p1$, ... , $3p3$.
To show that the relation p is reflexive, you must show that all elements in $X$ are in relation with themselves, i.e. $1\leqslant1$, $2\leqslant2$ and $3\leqslant3$ must be true. I hope it is clear to you that this is obviously true.
To show that p is transitive you must show that if $x\leqslant y$ and $y\leqslant z$ for $x,y,z$ in $X$ then $x\leqslant z$. For example $1\leqslant2$ and $2\leqslant3$ and $1\leqslant3$. However, You must show this for all elements in $X$. I think it is a good excercise for a first time to try all combinations to get a grasp of what a relation is.
To show that p is symmetric you must show that if $x\leqslant y$ then $y\leqslant x$. Hint: This relation is not symmetric. Can you find a counter-example?
p is not symmetric since (1,2) is there but (2,1) is not there.It is transitive since x<=y and y<=z implies x<=z.p is reflexive since x<=x.