Expression, table, and function

This overview explains the difference between written syntax, a truth table, and the Boolean function they describe. Different expressions can mean the same thing, while agreement on a few cases can conceal a real difference. For the input A zero and B one, the direct and expression returns zero because A is false. Now keep that same input fixed while evaluating a differently written expression. Changing expressions must not accidentally change the test input. Negating A gives one. Negating B gives zero. The inner or combines one and zero and returns one. This intermediate one is not the final answer, because the complete expression contains one more negation outside the parentheses. The outer negation changes that one to zero. The two expressions agree for zero one. We have compared the same complete input through two different instruction structures. This single match is useful, but it does not yet prove agreement for every input. An expression records syntax. Its Boolean function specifies the output for every complete assignment. A truth table lists that mapping in a declared variable order. Here the order is 00, 01, 10, 11. Compare A AND B with NOT of NOT A OR NOT B. Both output 0, 0, 0, 1. Different written syntax describes the same function. A OR B agrees at 00 and 11, but differs at both middle cases. At 01, AND returns zero while OR returns one. That single counterexample disproves their equivalence. all 4 cases are exhaustive for these two-input rules. Larger universes need an appropriate exact method. Work a second complete case with A 1, B 0. Keep this assignment fixed throughout the calculation. Each symbol retains the meaning already introduced; a new value does not create a new logical rule. Not A is zero and not B is one. Their inclusive or is one. Apply the outer not only after that inner result is known: not one is zero. Direct and of one and zero is also zero. Each intermediate negation has a different role from the final negation. This case establishes a specific contrast. The two expressions agree at 10. Do not omit the outer negation. Keep the input, intermediate values, and conclusion together when comparing it with the previous case. The calculated answer belongs to this complete assignment under the stated rule. For input one one, both inner negations are zero. Their or is zero, and the outer negation makes one. Direct and also returns one. The same sequence of operations works; we have not changed the meaning of negation between examples. The remaining inputs zero zero and one zero both give zero under each expression. All four rows agree. Because two Boolean inputs have exactly four assignments, this is an exhaustive equivalence check for this example, not a sample of a larger unstated universe. Compare and with or instead. They agree at zero zero and one one, so those two tests alone look reassuring. At zero one they disagree. One counterexample is sufficient to disprove equivalence. Establishing equivalence requires every relevant case or a sound proof covering them. One counterexample disproves equivalence. Complete agreement needs an exact check over the declared universe.