Exercise Set 2.1. Notes: is equivalent to AND ; both statements must be true for the statement to be true.
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1 Exercise Set ) Let p be the statement DATAENDFLAG is off, q the statement ERROR equals 0, and r the statement SUM is less than 1,000. Express the following sentences in symbolic notation. Notes: is equivalent to AND ; both statements must be true for the statement to be true. Symbol ~ is equivalent to NOT; the opposite of the statement. This sign changes an AND equivalent to an OR equivalent ; Symbol is a material conditional statement means if then ; example: If p then q, If you are taller than 5 8, you re tall. Symbol represents a Exclusive OR Statement and is only true when one of the statements are true and the other false, but never when both have the same value. a. DATAENDFLAG is off, ERROR equals 0, and SUM is less than 1,000. A: P^Q^R b. DATAENDFLAG is off but ERROR is not equal to 0. A: P^~Q c. DATAENDFLAG is off; however, ERROR is not 0 or SUM is greater than or equal to 1,000. A: P ~(Q^R) d. DATAENDFLAG is on and ERROR equals 0 but SUM is greater than or equal to 1,000. A: ~P^Q^~R e. Either DATAENDFLAG is on or it is the case that both ERROR equals 0 and SUM is less than 1,000. A: ~P (Q^R) 49) (p q) ( p q) ( q p) ( q p) by Commutative Law Ordaz 1
2 q (p p) by Distributive Laws ( Factor out NOT q) q c by Negation Laws ( (P^~P) negate to a contradiction) q by Identity Laws Therefore, (p q) ( p q) q. Section 3.3 Question #10 This exercise refers to Example Determine whether each of the following statements is true or false. a) students S, a desert D such that S chose D. b) students S, a salad T such that S chose T. c) a desert D such that students S, S chose D. d) a beverage B such that students D, D chose B. e) an item I such that students S, S did not choose I. f) a station Z such that students S, an item I such that S chose I from Z. a) For all students, there exists a dessert such that every student chose a dessert. - For all students there is a dessert that they chose. - True; every student had pie and Tim also had cake. b) For all students, there exists a salad such that every student chose. Ordaz 2
3 - For all students there is a salad that they chose. - False; Yuen did not choose a salad and Tim and Uta chose different salads. c) There exists a dessert such that for all students, the students chose this dessert. - There exists a desert that all students chose. -True; each student had a piece of pie. d) There exists a beverage such that for all students, the students chose this beverage. - There exists a beverage that all students chose. - False; Uta and Tim both chose Milk but Tim also had soda and Yeun was the only one to have soda. There was not one drink that all three had. e) There exists an item such that for all students, not one student chose this item. - There exists an item that all students did not choose. - False; each item was chosen at least once by one of the students. f) There exists a station such that for all students, there exists an item such that every student chose from the station. -There exists a station for which all students chose an item from. - True; All students chose items from Main courses, desserts, and beverages, the only station that does not apply is salads because Yeun was the only to not choose a salad. Section 3.3 Question #40 In informal speech most sentences of the form There is every are intended to be understood as meaning even though the existential quantifier there is comes before the universal quantifier every. Note that this interpretation applies to the following well-known sentences. rewrite using quantifiers and variables. a) There is a sucker born every minute. b) There is a time for every purpose under heaven. Ordaz 3
4 a) minutes M, a sucker S such that S is born in minute M. b) purpose P, a time T such that P is in time T. Theorems and Definitions -This set of problems were a set of multiply-quantified statements which are statements that contains more than one quantifier. -These statements contains informal versions of both the existential quantifier there is and the universal quantifier every. -In a statement containing both and, changing the order of the quantifiers tends to change the meaning of the statement. Ordaz 4
5 De Morgan s Equation for Success: Math and Logic The moving power of mathematical invention is not reasoning but imagination. Augustus De Morgan Augustus De Morgan is a mathematician who pioneered the connection between math and logic. He contributed many things to the field of mathematics and his many publications in England cover a variety of topics, including Algebra and Calculus, but the area he is most well known for is logic. His specific accomplishments within the field of logic are connecting math and logic, as well as theory of syllogism. His heritage and up bringing greatly influenced his role in the math realm. Augustus De Morgan was the fifth child of his family. His mother was a descendant of John Dodson, a recognized figure in the area of Anti-logarithms, which sheds light on Augustus passion for mathematics. He grew up in various private schools, but from an early age he showed significant interest and curiosity with mathematics. By the age of fourteen he had developed a clear talent for the subject and three years later entered Trinity College, Cambridge. It was at this institution that his interest in logic was cultivated by one of his professors, and lifelong friend, William Whewell (3). De Morgans love for math soon became even more passionate when he started recognizing math and logic should be linked. He continued his education at Cambridge and graduated with his BA in He did not receive his MA or a fellowship though because he refused to sign the theological test due to religious convictions (3). De Morgan then went on to become a well known and well liked Ordaz 5
6 teacher at University College London for the next thirty years, where he greatly influenced future mathematicians such as Isaac Todhunter, E. J. Routh, and J. J. Sylvester (3). While this aspect of his life is one most people will never accomplish in a lifetime, this is not all Augustus accomplished. De Morgan developed a passion for logic early in life. He especially put emphasis on logical training within math. He believed there were aspects to math that could not be solved or fully comprehended with out working directly with logic (1). De Morgan was associated with other great minds such as Sir W. R. Hamilton and George Boole for co-discovering the core principle of the quantifiable predicate. After De Morgan had a paper published, there arose conflict regarding this new term. This was the first reference to the term quantifiable predicate and therefore the credit went to De Morgan. Although there was brief controversy over who in fact coined this new term, all three colleagues are held as equal contributors. There was an additional controversy for the basis of De Morgan s logic system which was similar to Hamilton's. De Morgan had eight different aspects to his specific findings, and these propositions he used to form his logic system were from the foundation of the quantifiable predicate, not from Hamilton, though they were similar in thought process (1). These laws of logic are what lead to his commonly known name in the math community. Augustus is most recognized by name in the math community in regards to De Morgans Laws of logic when referring to theories of syllogism. His laws specifically explain that negating an AND statement changes the statement to an OR statement, and vice versa. This law has been instrumental in modern computer technology with software programming (1, 4, 5). These laws are also most commonly used in modern Ordaz 6
7 proof theory. Aristotle originally used logic syllogism, being the only one of his time to make such sound profound findings. His theory has since has been debated on a very small level, the only discrepancy being Aristotle sometimes contradicted himself using his syllogisms or was too vague. De Morgan took Aristotle s original process and refined it to be more specific with little to no room for contradiction (2). While there are many different aspects to the field of mathematics, Augustus De Morgan contributed to crucial aspects of modern technology and math. With out his brilliance, computer software might look very different today. The bridge between math and logic have taken both fields to whole new levels of understanding and comprehension, a place that could not have be achieved with out the help of Augustus De Morgan. Ordaz 7
8 Works Cited 1."Augustus De Morgan." Augustus De Morgan. N.p., n.d. Web. 8 Mar "Augustus De Morgan." Wikipedia. Wikimedia Foundation, n.d. Web. 11 Mar Brown, Scott H. "Applied Probability Trust." The Life and Work of Augustus De Morgan (2006): 4-9. Web. 8 Mar "Syllogism." Wikipedia. Wikimedia Foundation, n.d. Web. 13 Mar "Who Was Augustus De Morgan?" Who Was Augustus De Morgan? N.p., n.d. Web. 12 Mar Ordaz 8
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