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മലയാളം

സത്യപ്പട്ടിക നിർമ്മിക്കുക

Truth table of a Boolean expression in up to 6 variables, with the canonical sum of products, product of sums, minterms and maxterms.

പുതുക്കിയത് പരിശോധിച്ച ഉദാഹരണങ്ങൾ: 8

Operators: not (! ~ ¬ or a trailing '), and (& ·), nand, xor (^ ⊕), or (| +), nor, implies (->), iff (<->). Constants 0 and 1.
പരീക്ഷിക്കുക
Canonical sum of products
A'·B'·C' + A'·B·C' + A·B'·C' + A·B·C' + A·B·C
Canonical sum of products: A'·B'·C' + A'·B·C' + A·B'·C' + A·B·C' + A·B·C
Canonical product of sums
(A + B + C')·(A + B' + C')·(A' + B + C')
Minterms (rows that are 1)
Σm(0, 2, 4, 6, 7)
Maxterms (rows that are 0)
ΠM(1, 3, 5)
Rows that are true
5
Classification
Contingent

The expression is true in 5 of 8 rows (minterms 0, 2, 4, 6, 7), so its value depends on the inputs.

Columns of the truth table (row 0 on the left)

A
0
0
0
0
1
1
1
1
B
0
0
1
1
0
0
1
1
C
0
1
0
1
0
1
0
1
F
1
0
1
0
1
0
1
1
സത്യപ്പട്ടിക വരികൾ: 8
RowABCF
00001
10010
20101
30110
41001
51010
61101
71111
കണക്കാക്കുന്ന വിധം S

  1. Read with operator precedence

    (A∧B)∨¬C\left(A \land B\right) \lor \lnot C

    not binds tightest, then and/nand, xor, or/nor, implies (grouped from the right), and iff last.

  2. Evaluate all 8 rows

    Each row sets A, B, C to one combination of 0s and 1s, counting in binary with A as the most significant bit; the row number is the minterm index.

  3. Minterms and maxterms

    F=∑m(0,2,4,6,7)=∏M(1,3,5)F = \sum m(0, 2, 4, 6, 7) = \prod M(1, 3, 5)

  4. Canonical sum of products

    F=A‾B‾C‾+A‾BC‾+AB‾C‾+ABC‾+ABCF = \overline{A} \overline{B} \overline{C} + \overline{A} B \overline{C} + A \overline{B} \overline{C} + A B \overline{C} + A B C

    One product term for each row where F = 1; a bar means the variable is 0 in that row.

  5. Canonical product of sums

    F=(A+B+C‾)(A+B‾+C‾)(A‾+B+C‾)F = (A + B + \overline{C})(A + \overline{B} + \overline{C})(\overline{A} + B + \overline{C})

    One sum term for each row where F = 0; each term is 0 only in its own row.

സത്യപ്പട്ടിക നിർമ്മിക്കുക സംബന്ധിച്ച്

A truth table lists the value of a Boolean expression for every combination of its inputs: 2ⁿ rows for n variables, so 8 rows for A, B and C. Each row where the expression is 1 is a minterm, and the OR of those minterms is the canonical sum of products; each row where it is 0 is a maxterm, and the AND of the maxterms is the canonical product of sums. Both describe the same function.

Digital logic and computer science students use it to check circuit designs, simplify conditions and verify logic laws. The default, (A and B) or not C, is true in 5 of 8 rows, minterms 0, 2, 4, 6 and 7, and false in rows 1, 3 and 5.

Up to 6 variables (64 rows) are accepted. Precedence runs not, and/nand, xor, or/nor, implies (grouped from the right), then iff; symbols such as !, &, | and ^ also work. An expression true in every row is a tautology, and one false in every row is a contradiction.

പരിഹരിച്ച ഉദാഹരണങ്ങൾ

A and B

Boolean expression
A and B
Canonical sum of products
A·B
Minterms (rows that are 1)
Σm(3)
Rows that are true
1
Classification
Contingent

പരിശോധനയുടെ ഉറവിടം: Python itertools.product over (A, B): only A=1, B=1 is true

A xor B

Boolean expression
A xor B
Canonical sum of products
A'·B + A·B'
Minterms (rows that are 1)
Σm(1, 2)
Canonical product of sums
(A + B)·(A' + B')

പരിശോധനയുടെ ഉറവിടം: Python itertools.product with a != b

A implies B

Boolean expression
A -> B
Minterms (rows that are 1)
Σm(0, 1, 3)
Maxterms (rows that are 0)
ΠM(2)
Canonical product of sums
(A' + B)

പരിശോധനയുടെ ഉറവിടം: Python itertools.product with (not a) or b

Tautology: A or not A (edge case)

Boolean expression
A or not A
Classification
Tautology
Canonical product of sums
1
Rows that are true
2

പരിശോധനയുടെ ഉറവിടം: Law of excluded middle; Python check over A in (0, 1)

ചോദ്യങ്ങൾ

How do you make a truth table?

List every combination of the inputs by counting in binary: 2 variables give 4 rows, 3 give 8, and 6 give 64. Then evaluate the expression in each row, working from the innermost operation outward. For (A and B) or not C, the row A = 1, B = 1, C = 1 gives (1 and 1) or 0 = 1, while A = 0, B = 0, C = 1 gives 0 or 0 = 0.

What are minterms and maxterms?

A minterm is an AND of every variable, each plain or negated, that is 1 in exactly one row; a maxterm is an OR of every variable that is 0 in exactly one row. Row numbers index them: for A, B and C, minterm 6 is A·B·C′ and maxterm 1 is (A + B + C′). A function is the OR of its minterms and, equally, the AND of its maxterms.

What is the difference between sum of products and product of sums?

A sum of products (SOP) ORs together AND terms, one for each row where the function is 1; a product of sums (POS) ANDs together OR terms, one for each row where it is 0. A xor B is A′·B + A·B′ as a sum of products and (A + B)·(A′ + B′) as a product of sums. Both are complete; the shorter one is whichever has fewer rows to cover.

What is a tautology in logic?

An expression that is true in every row of its truth table, whatever the inputs. A or not A, the law of excluded middle, is the simplest; (A → B) or (B → A) is another. The opposite, false in every row, is a contradiction, such as A and not A, and anything true in some rows but not others is called contingent.

When is A implies B true?

A → B is false only when A is true and B is false; in the other three rows it is true. Its truth table therefore has minterms 0, 1 and 3 and a single maxterm, 2, and it is equivalent to not A or B and to its contrapositive, not B → not A. A chain such as A → B → C groups from the right, as A → (B → C).

“സത്യപ്പട്ടിക നിർമ്മിക്കുക” എത്രത്തോളം കൃത്യമാണ്?

കൃത്യത നിങ്ങളുടെ ഇൻപുട്ടുകളെയും രീതിയുടെ അനുമാനങ്ങളെയും ആശ്രയിച്ചിരിക്കുന്നു. ദശാംശ ഗണിതം 50 സാർഥക അക്കങ്ങൾ ഉപയോഗിക്കുന്നു. എന്നാൽ അനുമാനക്കണക്കുകൾ, സംഖ്യാത്മക രീതികൾ, ഉറവിട ഡാറ്റ എന്നിവയ്ക്ക് കൃത്യത കുറവാകാം; പ്രദർശിപ്പിക്കുന്ന മൂല്യം റൗണ്ട് ചെയ്യുന്നത് ഈ പരിമിതികൾ നീക്കില്ല. സ്വതന്ത്ര ഉറവിടങ്ങളിലെ പരിഹാരങ്ങളുമായി പരിശോധിച്ച ഉദാഹരണങ്ങൾ: 8. ഉദാഹരണത്തിന്, “A and B” എന്നത് Python itertools.product over (A, B): only A=1, B=1 is true ഉപയോഗിച്ച് പരിശോധിക്കുന്നു.

ഈ രീതിയുടെ ഉറവിടം എന്താണ്?

Wolfram MathWorld — Truth Table; Wikipedia — Canonical normal form (minterms and maxterms).

ഈ കാൽക്കുലേറ്ററിനെക്കുറിച്ച്

F=∑F(m)=1m(sum of minterms)=∏F(M)=0M(product of maxterms)\begin{aligned} F &= \sum_{F(m)=1} m \quad \text{(sum of minterms)} \\[4pt] &= \prod_{F(M)=0} M \quad \text{(product of maxterms)} \end{aligned}

സ്രോതസ്സുകൾ

  1. Wolfram MathWorld — Truth Table
  2. Wikipedia — Canonical normal form (minterms and maxterms)

ആധാരങ്ങളുമായി പരിശോധിച്ചത്

ഈ കാൽക്കുലേറ്ററിൽ സ്വതന്ത്ര ആധാരങ്ങളിൽ നിന്ന് ഉത്തരങ്ങൾ ലഭിച്ച 8 പരിഹരിച്ച ഉദാഹരണങ്ങളുണ്ട്. ഇവ പരിശോധനാസമുച്ചയത്തിൽ പ്രവർത്തിക്കുന്നു; നിങ്ങൾക്ക് ഇവിടെയും പ്രവർത്തിപ്പിക്കാം.

ബന്ധപ്പെട്ട കാൽക്കുലേറ്ററുകൾ