Finding the present property

Question

For each relation, determine which of these properties are present: reflexivity, symmetry, antisymmetry, and transitivity:

I know the definitions of each of the properties but unclear as to how to apply them to each relation.

{(a,a), (a,b), (b,a), (c,c), (d,d)}

{(a,a), (a,b), (a,d), (b,a), (b,b), (b,d), (c c), (d,a), (d,b), (d,d)}

{(a,a), (a,b), (a,c), (a,d), (b,b), (b,d), (c,c), (c,d), (d,d)}

Does anyone know what I would need to do to solve this?

Answer 1

Okay let's look at the first one the rest solve yourself! (I'll use $a\sim b$ instead of $a\text{R}b$

$$ A = \{a,b,c,d\} ;\quad R =\{(a,a),(a,b),(b,a),(c,c),(d,d)\} \subset A\times A$$

Symetric: $x\sim y \Rightarrow y\sim x$

Now we see $a\sim b$ is in the set aswell as $b\sim a$, all the other elements $(a,a),(c,c),(d,d)$ are already symetric.
We know it's not an antisymetric relation

Transitivity: $x \sim y, y\sim z \Rightarrow x\sim z$

Ok here we got $b\sim a$ and $a\sim b$ but $b\not\sim b$ because $(b,b)\not \in R$ so it's is not transitiv.

Reflexiv: $x\sim x$

Same example as above $b\not\sim b $ because $(b,b)\not\in R$

Answer 2

Let me clearly state the definitions for you.

Consider a relation $R$ on a set $A$.

Then,

$R$ is reflexive means $\text{for all $x\in A$, $(x,x)\in R$}$.

$R$ is symmetric means $\text{for all $x\in A$ and $y\in A$, $(x,y)\in R\implies (y,x)\in R$}$.

$R$ is transitive means $\text{for all $x\in A, y\in A$ and $z\in A$, $(x,y)\in R$ and $(y,z)\in R\implies (x,z)\in R$}$.

$R$ is antisymmetric means $\text{for all $x\in A$ and $y\in A$, $(x,y)\in R$ and $(y,x)\in R\implies x=y$}$.

Can you now identify for which of the sets the above properties hold?