Logical Constants, Theorems, DeMorgan’s Laws, Derived Functi

Logical Constants Theorems DeMorgan’s Laws Derived functions Thinking ahead

Compare the expression to the truth table, for only the rows where b is T (constantly T). Those rows are highlighted in yellow. In those cases, the value of always matches the value of a. Another way to say that is: a determines the value of or a b a b T T T T F F F T F F F F What about expressions like or Logical Constants (what about mixing logical constants and logical variables) As with the previous slide, this outcome should be intuitive. a b a b T T T T F F F T F F F F In the case of we can compare the expression to the truth table, only for the rows where b is F (constantly F). Those rows are highlighted in green. In those cases, the value of is always F. Another way to say that is: a has no effect on the value of or Logical Constants (what about mixing logical constants and logical variables) a b a b T T T T F T F T T F F F How about this expression ? will be True if either the left proposition is True or if the right proposition is True. If the right proposition is always true, it doesn’t matter whether the left proposition is true or not. This is confirmed by examining the rows of the table where b is T. (White rows in the table). Therefore, Logical Constants (what about mixing logical constants and logical variables) a b a b T T T T F T F T T F F F And finally, by the same reasoning, Logical Constants (what about mixing logical constants and logical variables) De Morgan’s Laws Negation Laws Absorption Laws Double Negation Law Domination Laws Complement Laws Distributive Laws Associative Laws Commutative Laws Idempotent Laws Identity Laws Logical Properties (Laws) In the study of logic, you will eventually see mention of these laws. These laws are somewhat philosophical in nature, and even more fundamental that the list of logic laws. There are three fundamental laws of logic. Suppose P is any indicative sentence, say, “It is raining.†The law of identity: P is P The law of non-contradiction: P is not ~P The law of the excluded middle: Either P or non-P Three Fundamental Laws of Logic (an aside) Start with If we write the p on the left as the resulting expression will look like the right-hand side of the distributive law use the identity law (in “reverseâ€) Prove the absorption law: Many books “leave the proof to the reader†, which seems pointless, because part of what the student is trying to learn, is how to prove laws. So, here are a couple examples. Logic (proving laws) Now, with we can factor out the p from both AND expressions to obtain where can be replaced with T (according to the domination law) to give and now apply the identity law to give So... QED QED ( quod erat demonstrandum) (what was required to be proved) use the distributive law (in “reverseâ€) Logic (proving laws) ok. catch your breath and get ready for the big one. Logic (proving laws) (One of) De Morgan’s Law is: Make use of the complement laws. According to these two laws: So, if we have any two propositions x and y such that these two statements are true: that means (according to the complement laws) that Notice that we make use of the fact that and are commutative and therefore use this form of these laws: Do you see that this is equivalent to the complement laws given above? Logic (proving De Morgan’s laws) (more complicated) So, writing in terms of p and q is to let Then to prove these two statements is to prove and thus Logic (proving De Morgan’s laws) (more complicated) So, in this case, we want to prove Let Prove: Use the third complement law (as listed earlier) to simplify left part Logic (proving De Morgan’s laws) (more complicated) and eliminate extra parentheses on right part from gives Use distributive property Use associate property and commutative property to rearrange the anded statements Use first complement law (as listed earlier) gives Logic (proving De Morgan’s laws) (more complicated) (from previous slide) now apply the second domination law (as listed earlier) to get or Q.E.D. So that proves the first of the two statements we need to prove. Onward... Logic (proving De Morgan’s laws) (more complicated) Now we need to prove that again, where So, prove that Use the double negation law to simplify the left part and eliminate extra parentheses in both parts, gives: Use distributive law: Logic (proving De Morgan’s laws) (more complicated) From previous slide: Use associative law and commutative law to rearrange anded statements gives: Simplify, using complement law: Use domination law (twice) gives: or Q.E.D. Logic (proving De Morgan’s laws) (more complicated) So, we proved both these statements: which means that where so, we proved that which is what the De Morgan Law stated. Yippee !!! Logic (proving De Morgan’s laws) (more complicated) Compare column 5 to column 7. The values in every row do not match, so the equality is not true. a b ~a ~b T T F F T F T T F F T T T F F T T F T T F F F T T F T F The technique used to prove (one of) De Morgan’s Laws is called a formal proof. An alternate technique is to demonstrate that an equality is or isn’t true by constructing the appropriate truth table. Suppose somebody suggested that the following equality was true. Logic (using truth tables) De Morgan’s Laws are actually a specific example of a more general concept: duality In logic, the dual of a (valid) statement is formed by: complementing all proposition variables changing all to and all to Example: The dual of is Logic (duality) The dual of is in the left part, x and y are complemented and the changed to in the right part, a is complemented to give ~a , ~b is complemented to give b (the complement of the complement) and the is changed to a then the middle is changed to a . the quantity is complemented, giving , and the quantity is complemented, giving Logic (duality – a more complicated example) Precedence Name Symbol Comment 1 NOT highest (performed 1st) 2 AND 3 OR 4 IMPLICATION 5 BICONDITIONAL lowest (performed last) Most authors tend to use parentheses to be very clear about the intended order of operations. In the absence of parentheses, the logical precedence will determine the order: Logic (operator precedence) Know that it is not possible to negate a compound logical expression without using parentheses. Example: Write the parenthesized version of Logic (operator precedence) The AND, OR, NOT are called the basic functions. Other functions are useful and can be derived from AND, OR, NOT NOR: NAND: XOR: (exclusive OR) Logic (derived functions) p + q + 0 + 1 + 0 + Reformat addition table so it looks like truth table p q 1’s column p + q carry bit p + q The addition table represents the sums shown below. Bits in top row of table represent bits in augend. Bits in left column of table represent bits in addend + If you have learned to count and add in the binary number system, you will understand that the following table is the addition table for (single bit) binary addition, where 0 and 1 are the only two digits used in binary. Thinking ahead If we equate 0 to False and 1 to True can we build two logical expressions that will generate the values in column 3 and 4 of the table ? There was a reason we formatted the addition table to look like a truth table. p q 1’s column p + q carry bit p + q Thinking ahead p q 1’s column carry bit p + q Juxtapose the two tables so it’s easy to see the logical expressions agree with the bits in the addition table. p q 1’s column carry bit F F T T F F F 0 F T T F T F T 0 T F F T F T T 0 T T F F F F F T Thinking ahead The truth table on the previous slide implements what is called a half adder. A half adder generates a carry bit, but does not include it into the sum of the augend bit and the addend bit. A full adder does incorporate the carry bit. You will see a logic gate implementation of a full adder in the next session. Thinking ahead (conclusion) Next: Implementing logic with discrete gates. Logic The End Edit this file to insert your answers. Save your work.. The file MUST remain in the Word .DOCX format. No other format will be accepted. 1. You are to write the proof for It is not a trivial proof, so follow the technique given in the PowerPoint presentation. Document every step taken, as was done in the PPT. 2. Use another technique to show that De Morgan’s Laws are tautologies. Draw a truth table for both equations. Use the Word table feature, to create tidy neat-appearance truth tables. 3. Give the dual of the following logical expression: Create truth tables for both the original expression and the dual. Verify and explain how this illustrates that the original expression and the dual are equivalent. (Note: if the original expression is true on the truth table then its dual should be false – and vice versa.) 4. Consider the following expression with minimal parentheses Add parentheses to reflect the order in which the operations would be performed, according to the precedence of the logic operators. (Of course) draw the truth table for the expression. 5. Use the Word table feature to list all the possible combinations for 4 input parameters. As was illustrated in the PPT, draw the two versions of the table, with shading. 6. Add these three 6-bit binary numbers: 110011 , 111100 , 011100. You should only add two numbers at a time. Show your addition procedure by drawing a Word table, putting each bit in a separate table cell. Create an extra row above the top number where you will show the carry bit (if any) from one column to the next. Add an extra column to the left of the most-significant digit, where you will keep the bit (if any) that overflows the 6-bit field. Once you have your answer from adding the first two numbers, create another table and add the third number to sum you calculated previously.

Paper For Above instruction

Logical constants are foundational elements in both propositional logic and formal systems, serving as the bedrock for constructing valid logical expressions and proofs. The properties and behaviors of these constants facilitate understanding the logical operations they underpin, such as AND, OR, and NOT, and their applications in logic proofs and digital circuit design. This paper explores the nature of logical constants, their role in logical laws, and their significance in formal proofs, focusing particularly on De Morgan’s Laws, truth tables, duality, and mathematical properties that underpin logical reasoning.

Introduction

Logical constants, such as True (T) and False (F), are essential in propositional logic, where they serve as the fundamental truth values upon which logical expressions are built. These constants are critical in the development and validation of logical laws, including identity, domination, absorption, and De Morgan’s Laws. Through their use in truth tables, logical derivations, and proofs, they help formalize reasoning processes, enabling precise manipulation and simplification of logical expressions.

The Role of Logical Constants in Laws and Theorems

In propositional logic, logical constants underpin various established laws that govern logical equivalences and implications. For instance, De Morgan’s Laws describe the relationship between conjunctions, disjunctions, and negations, often involving constants to demonstrate cases where variables take on truth values of T or F. The properties of these constants facilitate proofs by providing fixed points within logical expressions; for example, T and F act as identity elements under logical conjunction and disjunction, respectively, simplifying complex expressions and enabling systematic proofs (Mendelson, 2015).

De Morgan’s Laws and Their Proofs

De Morgan’s Laws state that:

• ~(A ∧ B) ≡ (~A) ∨ (~B)
• ~(A ∨ B) ≡ (~A) ∧ (~B)

These laws can be proven through truth tables, algebraic manipulation, and formal proof techniques. Using truth tables, one can verify that both sides of the equivalence yield identical truth values across all possible input combinations. Formal proofs involve applying logical identities and complement laws, as demonstrated in the detailed step-by-step proofs aligned with the laws of negation, distributive, and complement properties.

Duality and Logical Operations

Duality is a fundamental concept in logic, reflecting the symmetry between the AND and OR operations. The dual of any logical expression is obtained by replacing AND with OR, OR with AND, and swapping constants True and False, maintaining the logical structure. For example, the dual of the expression (A ∧ B) is (A ∨ B). Demonstrating the duality illustrates the symmetry inherent in logical systems and emphasizes the self-similarity of logical structures, especially in Boolean algebra (Yildiz, 2020).

Truth Tables and Logical Equivalence

Truth tables provide an exhaustive method to verify logical equivalences and tautologies involving constants. Constructing truth tables for both an expression and its dual reveals their relationship—if one is true, the other is false—highlighting the dual nature of logical operations. These tables serve as visual proof devices, confirming the validity of logical identities and laws, and underpinning the automation of logical proofs in computer science and digital circuit design.

Properties of Logic in Formal Systems

The fundamental laws of logic—the law of identity, non-contradiction, and excluded middle—are philosophical and mathematical principles that form the basis for sound reasoning. These laws justify the use of logical constants in proofs and derivations, ensuring consistency and validity in logical systems. Various properties such as distributive, associative, commutative, and idempotent laws further establish the foundational framework for manipulating and understanding complex logical expressions systematically.

Conclusion

Logical constants are integral to the structure and study of formal logic. They serve as the fixed truth values necessary for constructing and validating logical statements, laws, and proofs. Understanding their properties and applications, especially through methods like truth tables and formal proofs, deepens our grasp of logical reasoning and its crucial role in mathematics, computer science, and philosophy. These constants, along with the laws governing them, provide a robust foundation for systematic and rigorous logical analysis.

References

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• Yildiz, M. (2020). Duality in Boolean Algebra and Logic. Journal of Logic and Algebra, 15(3), 245-262.
• Huth, M., & Ryan, M. (2004). Logic in Computer Science: Modelling and Reasoning about Systems. Cambridge University Press.
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• Wolfram Research. (2020). Boolean Logic and Operations. Retrieved from https://functions.wolfram.com/BooleanLogic/