How to make formulas in mathematical logic?

Question

Mathematics

Hello! Please explain how to write formulas in mathematical logic. How it's done? I have an understanding of what conjunction, disjunction, equivalence, inversion and implication are. I don't know how to write formulas. Here's an example: to express equivalence through negation and conjunction; express equivalence through negation and implication. How to do it?

Answer 1

First, learn to put everything into words.
What is "equivalence"? This means that both operands (A and B) are equal, that is, either they are both equal to one, or both are equal to zero.
Both operands are equal to one: A∧B
Both operands are equal to zero :
¬A∧¬B
))
It remains to make a truth table and check

A | B | ¬(¬(A∧B) ∧ ¬(¬A∧¬B))
0 | 0 |           1
0 | 1 |           0
1 | 0 |           0
1 | 1 |           1

In the second case, the statement will be "from A follows B and from B follows A".
(A→B) ∧ (B→A)
It remains to get rid of the conjunction
¬((A→B) → ¬(B→A))

Answer 2

The most universal option is to compile a truth table, and use it to compile SKNF or SDNF