Question

Let a relation R = (x, -x)|x^2 = x (mod2x), whereby x element of positive integer. Determine whether R is reflexive, symmetric, anti-symmetric or transitive.

Answer 1

Answer on Question #63917 – Math – Discrete Mathematics

Question

Let a relation $R = (x, -x)|x^2 = x \pmod{2x}$, whereby $x$ element of positive integer. Determine whether $R$ is reflexive, symmetric, anti-symmetric or transitive.

Solution

Here $x^{2} = x (\mod 2x)$ and $(x, -x) \in R$.

If $x^{2} = x (\mod 2x)$, then $x^{2} = x + 2kx = (2k + 1)x$,

$$x = 2k + 1.$$

1) Is $R$ reflexive?

By a definition of a reflexive relation, all the pairs of $(x, x)$ must be in $R$.

But only the pairs $(x, -x)$ exist in $R$.

The only possible case is $x = 0$, but 0 is not a positive number.

Thus, a relation $\underline{R}$ is not reflexive.

2) Is $R$ symmetric?

By a definition of a symmetric relation, if $(x, y)$ are in $R$, then $(y, x)$ are in $R$ too.

By a definition of $R$, only pairs $(x, -x)$ are in $R$, where $x$ is a positive integer.

Pairs $(-x, x)$ may be regarded only when $x = 0$, but 0 is not a positive number.

Thus, a relation $\underline{R}$ is not symmetric.

3) Is $R$ anti-symmetric?

By a definition of an anti-symmetric relation, if $(x, y)$ are in $R$ and $(y, x)$ are in $R$, then $x = y$.

If $(x, -x)$ are in $R$ and $(-x, x)$ are in $R$, then $x = -x$, that is, $x = 0$.

The conclusion is true in this problem.

Therefore, a relation $R$ is anti-symmetric.

4) Is $R$ transitive?

By a definition of a transitive relation, the following property holds true:

$$(x, y), (y, z) \in R \to (x, z) \in R.$$

By a definition of $R$, only pairs $(x, -x), (y, -y)$ are in $R$.

Using the two previous definitions one conclude

$$- x = y \text{ and } - y = z \leftrightarrow - x = - z \leftrightarrow x = z.$$

Thus, a relation $R$ is transitive.

Answer:

1) nonreflexive; 2) nonsymmetric; 3) anti-symmetric; 4) transitive.

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