Build a combined rule negation table one case at a time

A combined rule negation table checks what happens when NOT applies to a whole expression. In this original paper worksheet, A and B each have a declared true-or-false value. The opposite of A AND B is true whenever at least one input is false. Requiring both inputs to be false would describe a narrower condition.

We will write all four input pairs, evaluate the original expression, and flip its complete result. Then we will check a rewritten expression against that result in every row. These are invented formula inputs, not an actual eligibility policy, software filter or instruction to take an action.

Define the three operations before filling a row

For this worksheet, A AND B is true only when A is true and B is true. A OR B uses inclusive OR: it is true when at least one input is true, including the case where both are true. Keep those definitions beside the page.

NOT flips the value of the expression it covers. NOT true is false, and NOT false is true. Parentheses identify the whole expression being flipped. In NOT(A AND B), first evaluate the expression inside the parentheses, then negate that one result.

A and B are independent input positions for this exercise. No rule says one must have the same value as the other. We therefore include false/false, false/true, true/false and true/true. Each position has two allowed values, and these four pairs exhaust their combinations.

Use the same row order throughout the worksheet. A row's position on the sheet does not determine its meaning; the two written input values do. If you later rearrange the rows, carry every result with its corresponding input pair.

Build the original AND column

For A false and B false, A AND B is false because the two inputs are not both true. With A false and B true, it is also false. One true input cannot satisfy our definition of AND.

For A true and B false, the result is false for the same stated reason: the false B prevents both inputs from being true. Finally, A true and B true gives a true result. The AND column, in our saved row order, reads false, false, false, true.

Write that complete column before negating it. This keeps the original result available for comparison. It also prevents a note such as “the false rows” from hiding whether you meant false inputs or a false combined result.

Negate the complete AND result

Now apply NOT to each completed result in the AND column. Its first three false entries become true. Its final true entry becomes false. Thus NOT(A AND B) reads true, true, true, false in the same four-row order.

Notice the false/true row. A AND B is false there, so its negation is true. The same happens in the true/false row. Those two mixed-input cases show why the negation does not require both inputs to be false.

The true/true row provides a different control. The original AND is true there, making the negation false. An expression claimed to negate A AND B must reject that row while accepting all three rows where the original result is false.

For every fully specified input pair, the original formula and its negation have opposite truth values. They cannot both be true in the same row, and they cannot both be false. Check that relation row by row rather than relying on the appearance of the written expressions.

Check the rewritten OR of the two negated inputs

Consider (NOT A) OR (NOT B). At false/false, NOT A is true and NOT B is true, so their inclusive OR is true. At false/true, the negated inputs are true and false, giving true again.

At true/false, the negated inputs are false and true, so the OR remains true. At true/true, both negated inputs are false, making the OR false. Its column is true, true, true, false.

This matches NOT(A AND B) in every allowed input pair. For this two-valued worksheet, NOT(A AND B) and (NOT A) OR (NOT B) therefore give the same result for every possible input. Both input negations and the change from AND to OR belong in the rewrite.

Preserve the four comparisons with the conclusion. A match in one chosen row would establish agreement only for that row. Here, completeness comes from checking the entire declared input set, rather than selecting examples that happen to support the proposed rewrite.

Expose the tempting AND rewrite with a counterexample

Suppose someone instead writes (NOT A) AND (NOT B). At false/false it is true, which agrees with NOT(A AND B). That single successful row can make the proposed rewrite look plausible.

Use false/true next. NOT A is true, while NOT B is false. Their AND is false. But A AND B was false, so NOT(A AND B) is true. The two expressions disagree on this fully specified input.

One such disagreement is enough to reject their equivalence. The true/false row supplies another disagreement, while true/true makes both expressions false. A record that kept only false/false and true/true would miss both cases that expose the error.

The incorrect expression still has a defined meaning: it requires neither input to be true. It is not meaningless or impossible to evaluate. It simply does not represent the complement of the original AND formula.

Repeat the method for an original OR formula

Return to the same four input pairs and evaluate A OR B using the saved inclusive definition. False/false gives false. False/true, true/false and true/true each give true. The complete OR column is false, true, true, true.

Negating that column gives true, false, false, false. This time only the false/false pair makes the negated expression true. The inclusive OR is false only when both inputs are false under our definitions.

Now evaluate (NOT A) AND (NOT B). Its results are true, false, false, false, matching NOT(A OR B) in all four cases. Here the correct rewrite changes OR to AND while negating each input.

Keep the two checked identities on separate lines: NOT(A AND B) equals (NOT A) OR (NOT B); NOT(A OR B) equals (NOT A) AND (NOT B). Swapping the original connective changes which rewritten connective belongs on the right.

Keep the scope of NOT visible

Compare NOT(A AND B) with (NOT A) AND B. In the first expression, NOT covers the whole AND result. In the second, it covers A alone, while B keeps its original value. They are different requests.

At false/false, the whole-formula negation is true. The second expression evaluates true AND false, which is false. That row alone distinguishes the two placements of NOT. Removing parentheses without an agreed reading rule can therefore conceal a changed expression.

You can avoid that ambiguity on the worksheet by keeping every negated compound inside parentheses and placing parentheses around each separately negated input. This is our written presentation convention, not a claim about the syntax or precedence rules of a particular application.

Applying NOT twice to the same formula restores its original value in this model. A true result becomes false and then true; a false result becomes true and then false. Keep the same parenthesized formula under both NOT operations when checking that statement.

Leave missing inputs and real instructions outside the result

An unknown A value is not one of the two allowed values already supplied for a row. Do not place it in the false row simply to finish the worksheet. Record the missing input and leave that actual case outside this fully specified example.

Nor does a false formula issue an opposite action. A real instruction saying to do something when both conditions hold may leave other cases unspecified. Our table evaluates truth values only. It supplies no additional instruction, permission, denial or default outcome for a real situation.

To use a real combined statement, establish what its inputs and connective mean in that setting. Do not silently transfer our inclusive OR convention into wording that has not defined it. This exercise proves the two rewrites under its stated definitions; it cannot settle an unresolved interpretation elsewhere.

Keep the two optical accounts within their evidence

The Eschenbach Mobilux account names a compact LED magnifier. Its sole image shows a complete handheld housing with ESCHENBACH/Germany print and a MagnifyingAids.com watermark. Those marks do not authenticate a maker or calibrate the listed lens size and magnification figures.

The INTBUYING 15X 127C account names a benchtop magnifier lamp. Four images show a complete white lens head, articulated arm, base and black adapter. Both accounts supply Home-only breadcrumbs, with no category assigned there. No optical power, operating, comfort, medical-use or delivered-contents result is verified.

Separate formula negation from other transformations

The extraneous-solution guide checks candidates against an original radical equation after squaring loses sign information. That original-equality check is different from constructing the complement of a fully defined Boolean formula.

The routine cue-replacement worksheet keeps an old row unchanged during a replacement pass. Our rows instead enumerate all input pairs for expressions. No newly written value feeds another row, and no cue replacement supplies the logical connective.

Keep any proposed purchase with its own information

If you separately consider an order on republeq.com and want to ask about USDT, begin with the exact offer and intended quantity before discussing payment. The truth values and row positions in this worksheet are invented paper inputs, not prices, token amounts or directions to send funds. A completed formula check establishes no available payment method or seller commitment. Read the choices and conditions actually presented for the intended transaction, and ask the responsible support contact to clarify essential missing information before deciding whether to proceed. If a cryptocurrency option is offered, establish the specified asset and network, requested amount, any stated validity period and the reference associating payment with that order. Do not infer a destination from an illustration, a previous transaction or a message about another proposed purchase. Resolve necessary item or supplied-contents questions separately from the payment discussion. Ask about applicable delivery, cancellation, return and refund conditions, including the denomination of an approved refund and relevant deductions. Keep each answer with the offer and quantity it concerns. A changed basket may require reviewing the resulting transaction information again. If an instruction has expired, obtain current information before making a transaction decision. This article promises no present USDT availability, conversion result, fee level, processing time, refund amount or delivery outcome. You can check every truth-table row without placing an order. Preserve any eventual acknowledgement and transaction record separately from the worksheet. Correct formula negation cannot confirm that a transaction was authorized, completed or reconciled. If transaction status remains uncertain, resolve it through the responsible merchant before considering another payment.

Retain the definitions and all four comparisons

A combined rule negation table is complete here when every allowed input pair has the original result, its flipped result and the proposed rewrite checked beside it. Keep the AND and OR examples attached to their definitions, with parentheses showing exactly what NOT covers.

The AND complement matches an OR of the negated inputs. The inclusive-OR complement matches an AND of the negated inputs. Those conclusions follow from the four recorded cases, while unknown inputs, ambiguous real wording and actual action instructions retain their own unresolved questions.