existential and universal successors

Question

I did not get it how universal successors are computed here.   As far as I know, universal successors are those states that can be reached from every state given in a set, 

Answer 1

I think the figure is incorrect!

Universal predecessors of {s3,s4} are those states where all successors are inside {s3,s4}. Checking the table, this is the case for states {s0,s3,s5} with the following explanations:

• suc(s0) = {} ⊆ {s3,s4}
• suc(s1) = {s2,s3} ⊈ {s3,s4}
• suc(s2) = {s1} ⊈ {s3,s4}
• suc(s3) = {s4} ⊆ {s3,s4}
• suc(s4) = {s2,s3} ⊈ {s3,s4}
• suc(s5) = {} ⊆ {s3,s4}

Universal successors of {s3,s4} are those states where all predecessors are inside {s3,s4}. Checking the table, this is the case for states {s4,s5}, but not for s1 as stated in the figure:

• pre(s0) = {s1} ⊈ {s3,s4}
• pre(s1) = {s2} ⊈ {s3,s4}
• pre(s2) = {s1,s2,s4} ⊈ {s3,s4}
• pre(s3) = {s1,s4} ⊈ {s3,s4}
• pre(s4) = {s3} ⊆ {s3,s4}
• pre(s5) = {s4} ⊆ {s3,s4}