Logic Circuits, Axioms for Basic Logic
(論理回路, 基本論理の公理化)
Discrete Mathematics I
5th lecture, October 25, 2019
https://www.sw.it.aoyama.ac.jp/2019/Math1/lecture5.html
Martin J. Dürst
© 200519 Martin
J. Dürst Aoyama Gakuin
University
Today's Schedule
 Schedule for next few weeks
 Summary of last lecture
 Last week's homework
 XOR, NAND, NOR, and logical circuits
 Axioms for basic logic
 Equivalence and implication
 This week's homework
Schedule for the Next Few Weeks
 October 25 (today): Logic Circuits, Axioms for Basic Logic
 November 1: No lectures (Aoyama Festival)
 November 8: Sets
 November 15: Predicate Logic, Quantifiers
 November 22: Applications of Predicate Logic
 November 29: Relations
Summary of Last Lecture
 There are many laws for logical operations. These laws can be used to
manipulate and simplify logical formulæ, and to prove additional laws.
 Most laws come in pairs that are related by the duality
principle.
 Each logical (Boolean) function can be represented in two normal forms:
 Disjunctive normal form: Disjunction of conjunction (of
negation) of variables
 Conjunctive normal form: Conjunction of disjunction (of
negation) of variables
 Normal forms can be simplified by symbol manipulation or by using a
Karnaugh map
Last Week's Homework
If you have a problem with accessing the course's web site or submitting your homework,
please contact me immediately by
email.
Grading is mostly manual and will take some time.
Homework Problem One
List all the possible Boolean functions of two variables A and
B in a table. Use one row for each Boolean function. Find the/a
simplest formula (using ¬, ∧, and ∨).
 Solution (logical operators that we will look at from now on are given in
[]):
A = F
B = F 
A = F
B = T 
A = T
B = F 
A = T
B = T 

F 
F 
F 
F 
F 
F 
F 
F 
T 
A ∧ B 
F 
F 
T 
F 
A ∧ ¬B 
F 
F 
T 
T 
A 
F 
T 
F 
F 
¬A ∧ B 
F 
T 
F 
T 
B 
F 
T 
T 
F 
¬A∧B ∨ A∧¬B;
(¬A∨¬B) ∧ (A∨B)
[A ⊕ B,
XOR] 
F 
T 
T 
T 
A ∨ B 
T 
F 
F 
F 
¬(A ∨ B) [A ⊽ B,
NOR] 
T 
F 
F 
T 
(A∨¬B) ∧ (¬A∨B);
¬A∧¬B ∨ A∧B [A ↔
B] 
T 
F 
T 
F 
¬ B 
T 
F 
T 
T 
A ∨ ¬B 
T 
T 
F 
F 
¬ A 
T 
T 
F 
T 
¬A ∨ B [A →
B] 
T 
T 
T 
F 
¬(A ∧ B) [A ⊼ B,
NAND] 
T 
T 
T 
T 
T 
Last Week's Homework: Problem Two
 Create a truth table for a Boolean function with four variables
(A, B, C, D).
 Decide on the result (truth value, T or F) for each row of the truth
table with a random function.
 As a random function, use e.g. a coin toss.
 Decide which side of the coin corresponds to which truth value
(e.g. Japanese 500yen coin: 500 side → true; flower side → false)
 Toss the coin as many times as necessary (16
times).
 Your Boolean function will be different from the Boolean function of all
other students.
 If your Boolean function is the same as that of another student, there
will be some deduction.
 Calculate the two normal forms and a simplified formula for your Boolean
function.
The solution is different for each student!
XOR, NAND, and NOR
Boolean functions other than AND, OR, and NOT used frequently in programs
and electronic circuits:
 XOR (exclusive or)
 defined as true if either A or
B is true, but not both
 written A xor B, A ⊕
B
 widely used in encryption/cryptography
 NAND
 defined as ¬(A ∧ B)
 written as NAND(A, B) or A ⊼
B
 widely used in electronic circuits and flash memory
 NOR
 defined as ¬(A ∨ B)
 written as NOR(A, B) or A ⊽
B
 widely used in electronic circuits and flash memory
Truth Tables for XOR, NAND, and NOR


XOR 
NAND 
NOR 
A 
B 
A ⊕ B 
A ⊼ B 
A ⊽ B 
F 
F 
F 
T 
T 
F 
T 
T 
T 
F 
T 
F 
T 
T 
F 
T 
T 
F 
F 
F 
Logic Circuits
 Boolean functions can be calculated (evaluated) using logic circuits
 The relationship between logic formulæ and electric (electronic)
circuits was established by Claude Shannon in
his 1938 master thesis
 The components of a logic circuit are called gates
 Gates are connected with wires
 The output of a gate is the input of (an)other gate(s)
 For logic circuits, 1 and 0 are often used in place of true and false
Gates Used in Logic Circuits
NOT gate 
AND gate 
OR gate 



XOR gate 
NAND gate 
NOR gate 



Example Logic Circuit: Half Adder
Half adder (to add two binary numbers, two half adders and an OR gate per
binary digit (bit) are necessary; this is called a full adder)
inputs 
outputs 
A 
B 
C (carry) 
S (sum) 
0 
0 
0 
0 
0 
1 
0 
1 
1 
0 
0 
1 
1 
1 
1 
0 
How to Write Logic Circuits
 Input from the left, output to the right
 Write variable name or formula for each input, output, and connection
 Make clear where crossing lines (wires) touch and where not
 AND, NAND, OR, and NOR gates may use more than two inputs
Examples:
AND(A, B, C) = A ∧ B ∧ C
NOR(A, B, C, D) = ¬(A ∨ B ∨ C ∨
D) ≠ A ⊽ B ⊽ C ⊽
D
OR(A) = A
NAND(A) = ¬A
Example Logic Circuit: Full Adder
inputs 
outputs 
A 
B 
C_{in} (carry in) 
C_{out} (carry out) 
S (sum) 
0 
0 
0 
0 
0 
0 
0 
1 
0 
1 
0 
1 
0 
0 
1 
0 
1 
1 
1 
0 
1 
0 
0 
0 
1 
1 
0 
1 
1 
0 
1 
1 
0 
1 
0 
1 
1 
1 
1 
1 
Formulæ:
C_{out} = A∧B ∨
C_{in}∧XOR(A, B)
S = XOR(XOR(A, B),
C_{in})
Property of NAND and NOR
Theorem: It is possible to implement all boolean functions using only NAND
[or only NOR]
Proof:
We know that we can write any boolean function using only ∧, ∨, and ¬
(using a normal form)
If we can write these three boolean operations using only NAND [or only
NOR], our proof is complete.
¬ A = NAND(A) [NOR(A)] (alternatively: A⊼A
[A⊽A])
A ∧ B = ¬¬(A ∧
B) = NAND(¬(A ∧
B)) = NAND(NAND(A,
B))
[= ¬¬(A ∧ B) =
¬(¬A ∨ ¬B) = NOR(¬A, ¬B) =
NOR(NOR(A), NOR(B))]
A ∨ B = ¬¬(A ∨
B) = ¬(¬A ∧
¬B) = NAND(¬A,
¬B) = NAND(NAND(A),
NAND(B))
[= ¬¬(A ∨
B) = NOR(¬(A ∨ B)) = NOR(NOR(A,
B))]
Q.E.D.
Comment: This is a constructive proof. It is often used for
theorems of the form "it is possible...". A constructive proof shows
how it is possible.
Comment: This property is important because it is easier to construct a
logical circuit where all the gates are the same (e.g. all NAND).
Axiomatization
 An axiom is a theorem that is assumed to be true, without proof.
 One goal of mathematics is to create rich, beautiful (and useful)
theories from very few axioms.
 In lecture 2, we introduced the Peano Axioms
for the natural numbers.
All provable properties of natural numbers can be proven from these very
few axioms.
 Can we find axioms for basic logic?
 Important properties for axioms:
 Consistent: No contradictions
 Complete: Possible to derive any true theorem
 Minimal: Impossible to remove any axiom
 Simple: Low number of axioms and operators, short
 Obvious to humans
Axiomatizations of Basic Logic
 Standard axioms:
A∨B=B∨A [+D];
A∧(B∨¬B)=A [+D];
A∨(B∧C)=(A∨B)∧(A∨C)
[+D]
 Huntington axioms (1904):
A∨F = A [+D];
A∨B=B∨A [+D];
A∨(B∧C)=(A∨B)∧(A∨C)
[+D]; A∨¬A=T [+D]
 Huntington axioms (1933):
A∨B=B∨A;
A∨(B∨C)=(A∨B)∨C;
¬(¬A∨B)∨¬(¬A∨¬B)=A
 Robbins axioms (proposed 1933, proved automatically 1996):
A∨B=B∨A;
A∨(B∨C)=(A∨B)∨C;
¬(¬(A∨B)∨¬(A∨¬B))=A
 Sheffer axioms (1913?):
(A⊼A)⊼(A⊼A)=A;
A⊼(B⊼(B⊼B))=A⊼A;
(A⊼(B⊼C))⊼(A⊼(B⊼C))=((B⊼B)⊼A)⊼((C⊼C)⊼A)
 Wolfram axiom (found automatically 2002):
(A⊼B)⊼C)⊼(A⊼((A⊼C)⊼A))=C
(for more axiomatizations, see http://www.cs.unm.edu/~veroff/BA/)
Comment: [+D] indicates that the dual is also an axiom
Evaluation of Axiomatizations

Consistent? 
Complete? 
Minimal? 
Simple? 
Obvious? 
Axioms 
Operators 
Overall Length 
Standard 
yes 
yes 
yes 
6 
3 
very long 
yes 
Huntington 1904 
yes 
yes 
yes 
8 
3 
very long 
yes 
Huntington 1933 
yes 
yes 
yes 
3 
2 
short 
no 
Robbins 
yes 
yes 
yes 
3 
2 
short 
no 
Sheffer 
yes 
yes 
yes 
3 
1 
long 
no 
Wolfram 
yes 
yes 
yes 
1 
1 
very short 
no 
Conclusion from Evaluation of Axiomatizations
 For natural number arithmetic (Peano axioms),
there are axiom systems that are both simple and obvious
 For plane geometry (Euclid's axioms),
there are axiom systems that are both simple and obvious
 For basic logic:
 There are axiom systems that are simple (but not obvious)
 There are axiom systems that are obvious (but not simple)
 There is no axiom system that is both simple and obvious
Logical Operations Important for Symbolic Logic
Truth Table


Equivalence 
Implication 
A 
B 
A ↔ B 
A → B 
F 
F 
T 
T 
F 
T 
F 
T 
T 
F 
F 
F 
T 
T 
T 
T 
 Equivalence: A ↔ B (read: A is
equivalent to B)
If and only if A and B have the same value (both T or
both F), then A ↔ B is T.
 Implication: A → B (read: A implies
B)
If A is T and B is F, then A →
B is F, otherwise A → B is T.
Example of Implication
A: "I will pass Discrete Mathematics I."
(antecedent)
B: "I will travel to Okinawa next Spring."
(consequent)
A → B: "Me passing Discrete Mathematics I implies
that I will travel to Okinawa next Spring."
or: "If I will pass Discrete Mathematics I, I will travel to Okinawa next
Spring."
Implication is True if Antecedent is False
 Assume you meet your friend in April 2020, receiving a small gift from
Okinawa. Your friend went to Okinawa.
 Question: Did your friend pass Discrete Mathematics I?
 Answer: We do not know. (The implication did not say
what happens if your friend does not pass Discrete Mathematics
I.)
 There are other cases where there are differences between everyday talk
and strict logic meaning of implication.
Laws of Implication and Equivalence
 Rewriting (removing) implication: A→B =
¬A∨B = ¬(A∧¬B)
 Rewriting (removing) equivalence: A↔B =
(A→B)∧(B→A) =
(A∧B)∨(¬A∧¬B)
 Transitive laws: ((A→B) ∧
(B→C)) ⇒ (A→C),
((A↔B) ∧ (B↔C)) ⇒
(A↔C)
 Reductio ad absurdum: A→¬A = ¬A
 Contraposition: A→B =
¬B→¬A
 Properties of equivalence: A↔B =
¬A↔¬B, ¬(A↔B) =
(A↔¬B)
 Properties of implication: T→A = A,
F→A = T, A→T = T, A→F =
¬A
Difference between → and ⇒
There is a difference between "→" and "⇒".
"→" is an operator of boolean/symbolic logic. "→" can be manipulated
like any other operator (e.g. ∧, ⋁, ¬,...).
"⇒" tells humans that we can replace formulæ that match the form
on its left side with formulæ that match its right side.
This is best visible in the transitive law for implication:
((A→B) ∧ (B→C)) ⇒
(A→C)
The same difference exists between "↔" and "=".
How to Use a Truth Table for a Proof
 Goal: Prove one of the absorbtion laws for ∨ and ∧: A∨A∧B
= A.
 Method 1:
 Create a truth table for both sides of the law.
 Compare the columns for both sides
(light green column and
light blue column).
 If the columns are equal, the law is proved
(always explicitly state this).
 Method 2:
 Create a truth table for both sides of the law.
 Add a column (yellow)
that combines both sides of the law with ↔.
 Check that the additional column contains all Ts (i.e. the formula is
a tautology)
(always explicitly state this).
A 
B 
A ∧ B 
A ∨
A∧B 
A∨A∧B
↔ A 
F 
F 
F 
F 
T 
F 
T 
F 
F 
T 
T 
F 
F 
T 
T 
T 
T 
T 
T 
T 
For both methods, add a sentence that says why the proof works.
Tautology and Contradiction
 A Boolean formula that is always true is called a
tautology.
 A Boolean formula that is always false is called a
contradiction.
 All laws are tautologies, but there are also tautologies that we don't
call laws.
(Example: T→A = ¬¬A)
This Week's Homework
Deadline: October 31, 2019 (Thursday), 19:00.
Format: A4 single page (using both sides is okay; NO cover
page), easily readable handwriting (NO printouts), name (kanji
and kana) and student number at the top right
Where to submit: Box in front of room O529 (building O, 5th floor)
(There are several boxes, please make sure to use the correct one.)
Problem 1: Express Boolean functions with NOR only:
For each of the 16 Boolean functions of two Boolean variables A
and B (same as problem 1 of last lecture), find the shortest formula
using only NOR. You can use NOR with any number of arguments ≧1, but you
cannot use T or F.
Hint: Start with simple formulæ using NOR and find out which functions they
represent.
Problem 2: Draw logic circuits of the following three Boolean formulæ:
 A ∨ ¬D ∧ C
 NAND(G, XOR(H, K), H)
 NOR(¬E, C) ∧ G
Additional Homework
Deadline: Thursday November 7, 2019, 19:00.
Format: A4 single page (using both sides is okay; NO cover
page; an additional page is okay if really necessary, but staple the pages
together at the top left corner), easily readable handwriting
(NO printouts), name (kanji and kana) and student number at the top right
Where to submit: Box in front of room O529 (building O, 5th floor)
(There are several boxes, please make sure to use the correct one.)
Problem 1: Prove/check the following laws using truth tables:
 Reductio ad absurdum (A→¬A = ¬A)
 Contraposition
((A→B)→(¬B→¬A))
 The associative law for conjunction
 One of De Morgan's laws
Problem 2: Prove transitivity of implication by formula manipulation:
Prove transitivity of implication (((A→B) ∧
(B→C)) ⇒ (A→C)) by formula
manipulation. For each step, indicate which law you used.
Hint: Show that ((A→B) ∧
(B→C)) → (A→C) is a tautology
by simplifying it to T.
Problem Three
(no need to submit/提出不要)
Create examples for implication and think about why the truth table for
implication is the way it is.
(含意の例を作って、含意の真理表と実例を比較して含意を身に着ける。)
Preparation for Next Lecture
(no need to submit)
In preparation for next week's lecture, using your high school
books/materials or other sources, research the following terms related to sets,
and write a definition and short explanation for each of them:
 Set
 Element
 Set union
 Set intersection
 Set difference
 Subset
 Proper subset
 Empty set
 Universal set
 Power set
(高校の本・資料や他の情報を活用して、上記の集合に関連する概念を調査し、定義と簡単な説明を書きなさい。)
Glossary
 grading
 採点 (作業)
 exclusive or
 排他的論理和
 encryption
 暗号化
 cryptography
 暗号学
 (electronic) circuit
 (電子) 回路
 logic circuit
 論理回路
 master thesis
 修士論文
 gate
 ゲート
 half adder
 半加算器
 bit (binary digit)
 ビット
 variable name
 変数名
 implement
 実装する
 theorem
 定理
 constructive proof
 構成的証明
 axiomatization
 公理化
 complete(ness)
 完全 (性)
 consisten(t/cy)
 一貫 (した・性)
 contradiction
 矛盾
 obvious(ness)
 明白な・自明性
 symbolic logic
 記号論理
 equivalence
 同値
 x is equivalent to y
 x と y が同値である
 implication
 含意
 x implies y
 x ならば y
 antecedent
 前件
 consequent
 後件
 pass (some course)
 (ある科目を) 合格する
 rewriting (removing) implication
 含意の除去
 rewriting (removing) equivalence
 同値の除去
 transitive laws
 推移律
 reductio ad absurdum
 背理法
 contraposition
 対偶
 properties of equivalence
 同値の性質
 properties of implication
 含意の性質
 tautology
 恒真式
