Predicate Properties
  1. If \forall x P(x) is true than \exists x P(x) is also true.
    \forall x P(x) \rightarrow \exists x P(x)
  2. If \exists x \forall y is true than \forall y \exists x is true
    \exists x \forall y \rightarrow \forall y \exists x   (One way theorem)
  3. \forall x \forall y P(x,y) \Leftrightarrow \forall y \forall x P(x,y)
  4. \exists x \exists y P(x,y) \Leftrightarrow \exists y \exists x P(x,y)

2Comments
Bharath Sathuri @bharathsathuri
25 Oct 2019 08:55 pm
can you explain the graph above?
Shamshad Hussain saifi @shamshadhussain
27 Oct 2019 01:54 am

@Sathuri Bharath Kumar Goud :Above graph is very useful to remember the predicate properties like

∀x∀yP(x,y)⇔∀y∀xP(x,y) true .It is two way theorem means true in both the direction

∋x∀yP(x,y)⇒∀y∋xP(x,y) true It is one way theorem means true in this direction only means not in reverse direction

∀y∋xP(x,y)⇒∋x∀yP(x,y) false

So basically this graph is very helpful to remember the all predicate properties