MHB Can you explain how the law of logic was used to reach this conclusion?

[CH1203]

New to set and graph theory and need help on how to approach these exercise questions:

For each of the following relations, state whether the relation is:
i) reflexive
ii) irreflexive
iii) symmetric
iv) anti-symmetric
v) transitive

Also state whether the relation is an equivalence or partial order relation. Give your reasoning.
a) x R y, if and only if x - y ≤ 3, where x and y $$\in$$ J
b) x R y, if and only if y / x $$\in$$ J, where x and y $$\in$$ N

I understand reflexive, irreflexive, symmetric, anti-symmetric and transitive, but I don't know how to work this out as I have only seen examples with matrixes which are visual...

 

[mathmari]

Re: HELP - Relations - State whether the relation is an equivalence or partial order relation

Consider the relation a). We have the following:

• A relation R is reflexive if for all x it holds that xRx.
  
  We have that $x-x=0\leq 3\Rightarrow xRx$. So, the relation is reflexive.
  
• A relation R is irreflexive if for no element x it holds that xRx.
  
  Since we have shown that the relation is reflexive, it cannot be irreflexive.
  
• A relation R is symmetric if for all x,y, xRy implies yRx.
  
  The relation is not symmetric, take for example $x=-5$ and $y=1$, then $x-y=-5-1=-6\leq 3$, but $y-x=1-(-5)=1+5=6\nleqslant 3$.
  
• A relation R is anti-symmetric if for all x,y, xRy and yRx implies $x=y$.
  
  Suppose that $x-y\leq 3$ and $y-x\leq 3$. These two inequalities hold for example when $x=1$ and $y=-2$:
  
  We have that $1-(-2)=1+2=3\leq 3$ and $-2-1=-3\leq 3$. So, if these inequalities hold it doesn't necessarily imply that $x=y$, so the relation is not anti-symmetric.
  
• A relation R is transitive if for all x,y,z, xRy and yRz implies xRz.
  
  Since xRy and yRz , we have that $x-y\leq 3$ and $y-z\leq 3$. Take for example $x=3, y=0, z=-3$. Then $x-z = 3-(-3)=3+3=6\nleqslant 3$.
  
  So, the relation is not transitive.

Partial orders and equivalence relations are both reflexive and transitive, but only equivalence relations are symmetric, while partial orders are anti-symmetric.

We have shown that the above relation is only reflexive. So, this relation is neither an equivalence nor a partial order relation

 

[solakis1]

Re: HELP - Relations - State whether the relation is an equivalence or partial order relation

Consider the relation a). We have the following:

• A relation R is reflexive if for all x it holds that xRx.
  
  We have that $x-x=0\leq 3\Rightarrow xRx$. So, the relation is reflexive.
• A relation R is irreflexive if for no element x it holds that xRx.
  
  Since we have shown that the relation is reflexive, it cannot be irreflexive.
• A relation R is symmetric if for all x,y, xRy implies yRx.
  
  The relation is not symmetric, take for example $x=-5$ and $y=1$, then $x-y=-5-1=-6\leq 3$, but $y-x=1-(-5)=1+5=6\nleqslant 3$.
• A relation R is anti-symmetric if for all x,y, xRy and yRx implies $x=y$.
  
  Suppose that $x-y\leq 3$ and $y-x\leq 3$. These two inequalities hold for example when $x=1$ and $y=-2$:
  
  We have that $1-(-2)=1+2=3\leq 3$ and $-2-1=-3\leq 3$. So, if these inequalities hold it doesn't necessarily mply that $x=y$, so the relation is not anti-symmetric imply that $x=y$, so the relation is not anti-symmetric.
• A relation R is transitive if for all x,y,z, xRy and yRz implies xRz.
  
  Since xRy and yRz , we have that $x-y\leq 3$ and $y-z\leq 3$. Take for example $x=3, y=0, z=-3$. Then $x-z = 3-(-3)=3+3=6\nleqslant 3$.
  
  So, the relation is not transitive.

Partial orders and equivalence relations are both reflexive and transitive, but only equivalence relations are symmetric, while partial orders are anti-symmetric.

We have shown that the above relation is only reflexive. So, this relation is neither an equivalence nor a partial order relation

So, if these inequalities hold it doesn't necessarily imply that $x=y$, so the relation is not anti-symmetric imply that $x=y$, so the relation is not anti-symmetric.

Which law of logic allow you to come to such a conclusion??

 

[HallsofIvy]

Re: HELP - Relations - State whether the relation is an equivalence or partial order relation

Which law of logic allow you to come to such a conclusion??

The examples he gave just before that sentence:
"We have that 1−(−2)=1+2=3≤3 and −2−1=−3≤3."