Problem $1.$

The following predicate logic formula is invalid:

AAx,EEy.$P(x,y)longrightarrowEEy,$AAx.$P(x,y)$

Which of the following are counter models for it$?$

The predicate $P(x,y)=\text{'}y*x=1\text{'}$ where the domain of discourse is $Q.$

The predicate $P(x,y)=\text{'}RP(x,y)={?}^{\text{'}}y*x=2RP(x,y)=yxy=xy\text{'}$ where the domain $of$ discourse $isR.$

The predicate $P(x,y)={?}^{\text{'}}y*x=2\text{'}$ where the domain $of$ discourse $isR$

without $0.$

The predicate $P(x,y)=\text{'}yxy=x\text{'}$ where the domain $of$ discourse $is$ the

set $of$ all binary strings, including the empty string.