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Here we have four basic relations and the properties that define the relations.
Reflexivity
For a relation to be reflexive, every element must be related to itself. So if „xRx“ is true, then R is a reflexive relation.
Ex: All real numbers are reflexive. 3R3, 5R5, 6R6. For the relevance of set theory, set inclusion is reflexive. Every set must include itself, so A⊆A, B⊆B, etc.
Symmetry
A relation is symmetric if the ordering of the pairs can be reversed. So if „xRy“ and „yRx“ are both true, then the relation R between them is symmetrical.
Ex: The equality relation is symmetrical. 2+2=4 is the same as 4=2+2.
Transitivity
A relation is transitive if whenever A is related to B, and B is related to C, then A is related to C.
Ex: „< " and ">“ are transitive. 3<4 and 4<5, so 3<5. Similarly, Ohio is within the United States, and the United States is within the world, so Ohio is within the world. For the relevance of set theory, set inclusion is transitive. If A⊆B and B⊆C, then A⊆C.
Antisymmetry
Ok, here’s where I get pretty terribly confused.
According to what I‘ve read, a relation is antisymmetrical if the relation between A and B, and B and A, is such that it implies A=B.
So, for example, set inclusion is antisymmetrical. If A⊆B, and B⊆A, then we must infer that in fact A and B have the same elements, and are therefore equal.
The problem
Now, according to the textbook I‘m using (Naive Set Theory by Paul Halmos), and ‚Innensenat nauforename mathmath red to ratz Neumann in harm and positiv click away.

Urdu….
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