Let S be a set R be a binary relation on S and x an element

Let S be a set, R be a binary relation on S, and x an element of S. Translate the following into a logical expression with the same meaning: the negation of the statement \"For all x in S, xRx.\"

Solution

Solution :

The negation of a relation R is the relation ‘notR’, for which x(notR)y is true

if and only if xRy is false in a given context I that contains XY.

The negation of a relation is often shown by the same symbol with a stroke through it.

[For example = and ]

So the negation of the statement is   \" There exists an x in S such that (x,x) R \"

OR we can write like this :

\"There exists an element x in the set S, such that x is not related to itself under R.\"