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It is only possible when the set has only one element.

S={a}

if the set has more than one element then the predicate will return false.

so maximum cardinality is 1.
2 votes
2 votes
for all x for all y (x=y)

This is true when for every element in the domain, x = y is true.

equality holds true when LHS = RHS means if x value is = to y value then its true.
 

And only if LHS = RHS,then the Predicate is TRUE. and that is possible when both LHS and RHS has same value. That is single element.

The moment we take 2 values they both should be same or both should be differrent.

Case 1 : If both are same then LHS  = RHS means same element repeated twice taken as single element in the domain (which is a set)

 

Case 2 : if both are differrent LHS $\neq$ RHS, that means there are 2 differrent elements in the domain and the predicate becomes false in this case.

So, The conclusion is When the Universe contains only 1 element,then the Predicate is always true.
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