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

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

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

solakis said:
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."
 
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