Basics of "Concept of relation" and "less than equal to"

Question

Let R be a relation defined on set of natural numbers N such that x is related to y iff x is less than equal to y i.e xRy iff x $\le$ y .Now I can understand this relation is reflexive and transitive but not symmetric but I want to ask whether (5,6) belongs to relation R ? If yes then it means 5 is less than equal to 6 as well as 5 is less than 6 simultaneously? So what is the proper mathematical meaning of " less than equal to"

Answer 1

The symbol $\leq $ means "less than or equal to" and is satisfied if "is less than" or "is equal to" is true.

Answer 2

In Mathematics the word "or" is inclusive.

$$ x \le y \iff (x<y ) or (x=y)$$

That means if x is less than y, we can say x is less than or equal y.

Also if x=y we can say x is less than or equal y.

Thus if one component is true, the compound statement will also be true.

Yes, (5,6) belongs to the relation because the component 5<6 is true.

Answer 3

This might help:

If $x\le 6$ is true, means that no necessarily $x=6$.

Or, If $x\le 6$ is true, means that no necessarily $x<6$.

Answer 4

The mathematical relation of "less than or equal to" ($\leq$) for any two numbers $a,b \in \mathbb{R}$ is defined as $a \leq b$ if and only if $a < b$ or $a = b$. Utilising logic here, we can see that the $\textbf{or}$ means that at least one of the statements is true, i.e either $a < b$ or $a = b$.