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

AI Thread Summary
The discussion focuses on analyzing specific relations to determine their properties, including reflexivity, symmetry, anti-symmetry, and transitivity. For the relation defined by x R y if x - y ≤ 3, it is established as reflexive but not symmetric, anti-symmetric, or transitive, thus classifying it as neither an equivalence nor a partial order relation. The reasoning involves applying logical principles, such as the definitions of each property and counterexamples to disprove others. The participants seek clarification on the logical laws that support these conclusions. The conversation emphasizes the importance of understanding these foundational concepts in set and graph theory.
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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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