Implication and logical equivalence

Two propositional operators look like reasoning operators.

symbol	read	write as
→	if then (implication)	-->
↔	if and only if (biconditional)	<->

Express the following claims.

If Bern is nice, then also Zürich is nice.
prop_logic ok_text Correct! fail_text Please try again! You must use → or ↔ or both. f_nodes 3 hide_expr B → Z ⇔

Exactly one of Bern and Geneva is nice.
prop_logic ok_text Correct! fail_text Please try again! You must use → or ↔ or both. f_nodes 4 hide_expr B ↔ !G ⇔

Zürich is nice but Bern is not. (Now you must use → or ↔ or both.)
prop_logic ok_text Correct! fail_text Please try again! You must use → or ↔ or both. f_ban ∧; f_nodes 4 hide_expr Z ∧ !B ⇔

The differences between ⇔ and ↔, or between ⇒ and →, are a long story. Let us just show with an example one difference. (You may modify the example and try again, but please be aware that here the tool computes everything modulo 21. This is yet another imperfection.)

modulo 21 ok_text Correct! 4x−7 = 13 ⇔ 4x = 20 ⇔ x = 5

modulo 21 ok_text Correct! fail_text When #<i#>x#</i#> = 0, both 4#<i#>x#</i#> − 7 = 13 and 4#<i#>x#</i#> = 20 yield #<b#>F#</b#>. Because they yield the same truth value, 4#<i#>x#</i#> − 7 = 13  ↔  4#<i#>x#</i#> = 20 yields #<b#>T#</b#>. However, #<i#>x#</i#> = 5 yields #<b#>F#</b#>. Therefore, (4#<i#>x#</i#> − 7 = 13  ↔  4#<i#>x#</i#> = 20)  ↔  #<i#>x#</i#> = 5 yields #<b#>T#</b#> ↔ #<b#>F#</b#> which is #<b#>F#</b#>. Just like (#<i#>P#</i#> ∧ #<i#>Q#</i#>) ∧ #<i#>R#</i#> means the same as #<i#>P#</i#> ∧ #<i#>Q#</i#> ∧ #<i#>R#</i#>, also (#<i#>P#</i#> ↔ #<i#>Q#</i#>) ↔ #<i#>R#</i#> means the same as #<i#>P#</i#> ↔ #<i#>Q#</i#> ↔ #<i#>R#</i#>. Please try adding the parentheses to the given answer! #<br#>#<br#>Also try adding the parentheses to the previous given answer (the one using ⇔). TT ⇔ 4x−7 = 13 ↔ 4x = 20 ↔ x = 5

The issue mentioned in the previous feedback can also be seen from expression trees. expression_tree
4x−7 = 13 ⇔ 4x = 20 ⇔ x = 5

expression_tree 4x−7 = 13 ↔ 4x = 20 ↔ x = 5