Discrete Structures Readings Check Section 21

Discrete Structuresreadings Check Section 21read Section 21 Pages 4

Discrete Structuresreadings Check Section 21read Section 21 Pages 4

Discrete Structures Readings Check section 2.1 Read Section 2.1, pages , and answer the questions below: Type in the answers below each question 1) What two types of sentences are not regarded as statements? (Can you name a third type?) 2) Under what conditions is the statement p ( q true? 3) What is the one case in which the statement p ( q is false? 4) Under what conditions is the statement p ( q true? 5) What connective is used in place of the word "but". 6) How many rows are in a truth table with three variables, like p, q, r? 7) Referring to example 2.6 and definition 2.1, what type of statement is p ( (p ( q) ? Discrete Structures Readings Check section 2.2 Read Section 2.2, pages , and answer the questions below: [ignore pages ] Type in the answers below each question 1) From table 2.6, give a statement that is logically equivalent to p ( q. 2) From table 2.7, give a statement that is logically equivalent to p ( q. 3) What are DeMorgan's Laws? [important] 4) What are the two Idempotent Laws? 5) What are the two Absorption Laws? [These will be very handy on homework and quiz.] 6) Using the Principle of Duality, write a logical equivalence based on the equivalence p ( (p ( T0 ? [page 59] 7) What is the relationship between the tautology (p ( q) ( (q ( p) and the tautology ((r ( s) ( (u ( v)) ( ((u ( v) ( (r ( s)) ? [see the substitution rules on p. 60 and example 2.10] 8) List both the contrapositive, and the converse of the implication statement (r ( s) ( u. Which of the two is equivalent to the original implication statement? [see pages ] Discrete Structures Readings Check section 2.3 Read Section 2.3, pages , , and answer the questions below:. 1) How many premises can an argument have? How many conclusions? 2) Is the argument presented in table 2.14 a valid argument? How do we know it is/isn't ? 3) Looking at page 69, if p ( q then what can we say about the statement p ( q ? 4) As explained at the top of page 70 what is the potential problem with using a truth table to establish the validity of an argument? 5) In the "tabular form" of an argument, what goes above the line? What goes below the line? 6) What rule of inference shows that the following argument is valid? "If x is an amanita muscaria then x is in the class agaricomycetes. If y is in the class agaricomycetes then y is in the Fungi kingdom. Therefore, amanita muscaria is in the Fungi kingdom." 7) What rule of inference establishes the validity of the following argument: (r ( s) ( w (w ((r ( s) 8) Is the following argument valid? (See page 74) r ( s s r 9) The argument in example 2.34 is invalid. Briefly summarize how this is shown (on page 83). Weekly Summary, Week 2, Chapter 2, Discrete Structures Type your answers under the questions given below, or, print out this sheet, use pen or pencil to fill it out, and email a scan or photo of it to the instructor. 1) a) Write out the contrapositive to the statement ((p ( (q ) ( ((p ( (q) b) Is the result in part a) a tautology? 2) Using the Principle of Duality, write out the dual of the logical equivalence: (p ( q) ( (p ( r) ( (p ( (q ( r) [Hint: you must first covert the statement on the right into a disjunction] 3) Write out a statement that gives exactly the same final result column (under the ?) as the following truth table: p q r ? ) a) How many rows (not including column headers) are required to make a truth table for a statement that contains 10 different letters (primitive statements)? b) How many different final result columns are possible in a truth table for a statement that contains 10 different letters? [Just set up the answer; do not reduce it to a single number.] [For example, a truth table for a statement with 2 variables, such as p and q, would contain 4 rows. Each row can be 0 or 1 in the final result column. Therefore, using the product rule from section 1.1, there would be 24 = 16 possible ways to fill out the final result column.] 5) Negate and (greatly) simplify the following statement: ((p ( (p ( q)) ( (((p ( (q). 6) Is the following argument valid or invalid? [((p ( s) ( ((t ( (s ( r)) ( ((q ( r) ( (p ( q ( t)] ( (r ( s) [There is a small bonus if you can precisely explain your answer.] 7) The following argument is invalid. Assign logical values (0 or 1) to the individual letters in such a way that all of the premises are true and the conclusion is false, thus proving that the argument is invalid. (P ( (Q R ( S W ( Q (R ( W (W ( P (((((((((( P ( S bonus: Write the following argument in symbolic notation.

Is it a valid argument? [Hint: for the primitive statements use the symbols a1 , a2 , a3 , b1 ,…, and also v1 v2 v3 .] "If Alvarez supports issue 1 and supports issue 2 and opposes issue 3 then I will vote for Alvarez. I will vote for Alvarez or Brown or Chang. Brown supports issue 3 but opposes issue 2. Chang agrees with Alvarez on issues 1 and 2. Chang disagrees with Brown on issues 2 and 3. Alvarez supports issue 1. If Chang supports issue 1 then Alvarez will oppose issue 3. If I vote for Alvarez then I will not vote for Chang. If Brown supports issue 1 then I will vote for Chang. Therefore, I will vote for Chang.

Paper For Above instruction

Discrete Structures are fundamental in understanding logical reasoning, set theory, and the principles of computing. The given assignment involves interpreting logical statements, truth tables, and inference rules which are essential topics in discrete mathematics.

Understanding Statements and Logical Connectives

First, it is crucial to distinguish between types of sentences and when they qualify as statements. Sentences that are questions, commands, or expressions of emotion are generally not regarded as statements because they do not assert a fact that can be evaluated as true or false. For example, commands like "Close the door" or questions like "Is it raining?" are not statements. There are also other non-statements such as wishful expressions or beliefs that are not definitively true or false, but traditionally, the main types are declarative sentences that make assertions.

The conditions under which the compound statement p ( q is true are when either p is false or q is true; graphically, it is false only when p is true and q is false. Conversely, when p is true or q is false, the disjunction p ( q remains true. The connective used in place of the word "but" is typically the logical connective "and" (conjunction). The truth table for conjunction shows that p ( q is true only when both p and q are true.

A truth table with three variables p, q, r contains 2^3 = 8 rows, representing all possible combinations of truth values for these variables. Such tables systematically list each combination to evaluate the truth of complex statements built from them.

Logical Equivalences and Laws

Referring to examples and definitions like 2.6 and 2.1, the statement p ( (p ( q) is a compound statement that is equivalent to p ( q under certain logical identities. These identities help simplify complex logical expressions. DeMorgan's Laws are fundamental in logic, allowing the transformation of logical expressions: the negation of a conjunction is equivalent to the disjunction of the negations, and vice versa.

The Idempotent Laws state that p ( p is equivalent to p, and q ( q is equivalent to q. The Absorption Laws help simplify expressions like p ( (p ( q) to just p, confirming the redundancy of repeated conditions.

The Principle of Duality states that every logical expression has a dual: interchanging conjunctions with disjunctions and selecting true and false constants accordingly results in an equivalent expression. For instance, the equivalence p ( (p ( T0 can be derived from these principles, showcasing duality.

Regarding tautologies such as (p ( q) ( (q ( p), their relationship with other tautologies can be appreciated through logical equivalences, highlighting how symmetry and the principle of duality underpin logical reasoning.

Implications and Argument Validity

Arguments with multiple premises and singular conclusions are common in proofs and reasoning. Validity hinges on whether the conclusion logically follows from the premises. An argument shown in table 2.14 can be evaluated using truth tables or inference rules to determine validity.

The rule of inference such as hypothetical syllogism or modus ponens often proves the validity of arguments, like the one with the premises involving classes and support issues mentioned. These rules establish the logical connection between premises and conclusion. For example, the rule of hypothetical syllogism, which states that from p implies q and q implies r, we can infer p implies r, is pertinent here.

The potential issues with truth tables, such as their impracticality with very large statements—like those containing 10 different primitive statements—are due to the exponential growth in the number of rows (2^n), which becomes computationally infeasible.

Complex Argument Analysis

Complex arguments, like the one supporting voting behaviors based on support and opposition to issues, can be symbolically represented using propositional logic. For example, let a1, a2, a3 represent support for issues 1, 2, and 3 respectively; b1, b2, b3 represent opposition; and v1, v2, v3 represent voting for individuals. The validity of such an argument can then be tested by constructing a logical formula and verifying if the conclusion necessarily follows from the premises, using methods such as truth tables or inference rules.

Conclusion

Discrete structures form the backbone of modern logic, computer science, and mathematics. Mastery of their topics, including logical equivalences, inference rules, and truth tables, is essential for analyzing complex arguments and understanding computational reasoning. These topics not only underpin theoretical computer science but also have practical implications in areas such as digital circuit design, algorithm development, and formal verification.

References

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