- Slides: 21
MATH 170 Project #9 Scarlitte Ponce Starlite Ponce Cristina Hernandez
Section 3. 3 #44 The Notation ∃! stands for the words “there exists a unique”. Thus for instance, “∃! x such that x is prime and x is even” means that there is one and only one even prime number. Which of the following statements are true and which are false? Explain. a. ∃! real # x such that ∀ real numbers y. xy=y b. ∃! integer x such that 1/x is an integer c. ∀real numbers x, ∃! real # y such that x+y=0
Notations ( make the question simpler to read) ∃!= “There exists a unique” Example: “∃! x such that x is prime and x is even”= there is one and only one even prime number ∀= “For all” ℝ= Real number, 1 1/2, √ 3, π ℤ= integers, -3. -1, 1, 9 ∈= an element of x∈y
Rewrite #44 Easier to read the question a. ∃! real # x such that ∀ real numbers y. xy=y a rewritten. There exists a unique real number x such that all real numbers y. x time y equals y. a rewitten. ∃! x∈ℝ, ∀ y∈ℝ. xy=y b. ∃! interger x such that 1/x is an integer b rewritten. There exists a unique real integer x such that 1 divided by x is an integer b rewritten. ∃! x∈ℤ, 1/x is an ℤ c. ∀real numbers x, ∃! real # y such that x+y=0 c rewritten. All all x real numbers, There exists y real number, such that x+y=0 c rewritten. ∀ x∈ℝ. , ∃! y∈ℝ, x+y+=0
Solve a. ∃! real # x such that ∀ real numbers y. xy=y Algebra Let x=1 xy=y (1)y=y Explanation x = there exists a unique real number=1 y= all real numbers plug in 1 for x y∈ℝ Hence, y is a real number and xy, (1)y, y=y. The statement is true.
Solve b. ∃! integer x such that 1/x is an integer Algebra Let x=1 1/x 1/1 1 Explanation x = there exists a unique real number=1 original problem plug in 1 for x 1∈ℤ Hence, 1 is an integer. Statement is true.
Solve c. ∀real numbers x, ∃! real # y such that x+y=0 Algebra x+y=0 y=-x unique number y= all real numbers negative x Explanation original problem subtract x y and x must be equal Hence, x is not equal to y because x is all real number and y is a unique number. Statement is false.
Section 2. 4 #23 Design a circuit to take input signals P, Q, and R and output a 1 if, and only if, all three of P, Q, and R have the same value
Explantion = And gate = ^ = Or gate= V = Not gate = ~ We use circuits to draw statements and we use them in daily life
Explantion sum of products: the sum of product is when you're given a table that has 1 and 0 the one 0<1 use products of sum if you have 0>1 us sum of products After you find out what sum to use, in this case we will use sum of products, you create a boolean function and you find P, Q, and R that have the number of 1. then you make P, Q, and R equal 1 in the table so, for example if P is 0, then you write it as ~P instead of P. After that you continue on with Q and R. In this case you end up with P^Q^R. in the next column it is ~P^~Q^~R. To connect this you add an or, so you end up with (P^Q^R)V(~P^~Q^~R) by sum of products.
Design a circuit to take input signals P, Q, and R and output a 1 if, and only if, all three of P, Q, and R have the same value Step 1 P 1 1 0 0 Q 1 1 0 0 R P⇔Q⇔R 1 1 (P^Q^R) 0 0 1 0 (P^Q^R)v(~P^ ~Q^ ~R) 0 0 1 (~P^ ~Q^ ~R) ⇔= “if and only if” Biconditional Statement we need to construct a truth table where P, Q, and R have all different values tested. when all P, Q, R are the same, the statement is equal to 1 but the statement is false or 0 when one or more are different among each other.
Step 2 P Q R 1 AND NOT AND Hence done.
Section 2. 4 #25 An alarm system has three different control panel in three different locations to enable the system, switches in at last two of the panels must be in the on position. If fewer than two are in the on position the system is disabled. design a circuit to control the switches.
An alarm system has three different control panel in three different locations to enable the system, switches in at last two of the panels must be in the on position. If fewer than two are in the on position the system is disabled. design a circuit to control the switches. PQR TTTT TTFT TFTT TFFF FTTT FTFF FFTF FFFF P^Q^ R P^Q^~R P^~Q^R ~P^Q^ R (P^Q^R)V(P^Q^~R)V(P^~Q^R)V(~P^Q^R)
STEP 2 Draw the Circuit (P^Q^R)V(P^Q^~R)V(P^~Q^R)V(~P^Q^R) P Q R a n d OR a n d A N D
2. 4 # 34 Show that the following logical equivalences hold for the pierce arrow, where P Q = ~(Pv. Q) a) ~P= P P b) PVQ= (P Q) c) P^Q= (P P) (Q Q) d) Write P→Q using pierce arrows only e) Write P⇔Q using pierce arrows only
Solve a) ~P= P P ~P= ~(Pv. P) De Morgan's laws ~P^~P Idempotent laws =~P b)PVQ= (P Q) ~(PVQ) ~(~PVQ) V (~PVQ) Double negative (PVQ) V (PVQ) Distribution PV(QVQ) Idempotent =PVQ
Solve c) P^Q= (P P) (Q Q) ~(PVP) ~(QVQ) De. Morgan’s ~P^~P ~Q^~Q Indempotent ~P ~Q Pierce Arrow ~(~Pv~Q) De. Morgan’s P^Q Final Answer. d) Write P→Q using pierce arrows only P→Q= ~Pv. Q Negate ~P Q=P^~Q let P be ~P ~(~P Q)= Use De. Morgan’s ~Pv. Q=P→Q Simplifed ~(~P Q) Final Answer
slove e)e) Write P⇔Q using pierce arrows only P→Q ~Pv. Q ~P Q=P^~Q ~P or ~Q ~(~P Q) ~(~(~Pv. Q) ~(P^~Q) Law ~Pv. Q ~(~P Q) Q→P Seperate into 2 ~Qv. P Negate ~Q P=Q^~P Orinal statement with ~(~Q P) Pierce Arrows ~(~(~Qv. P) De. Morgan’s Law ~(Q^~P) De. Morgan’s ~Qv. P De. Morgan’s Law ~(~Q P) Answer
Biography: C. S. Peirce (1839 -1914) Two gates that were not previously introduced were the NAND-gate and the NOR-gate. They are used to simplify circiuts. A NAND-gate is used like and AND-gate but is followed by a NOT-gate. A NOR-gate is like an OR-gate but is also followed by a NOT-gate. The notations for these gates are | (for NAND) and ↓ (for NOR). The notations for an NAND-gate, |, is called a Sheffer stroke after H. M. Sheffer and the notation for a NOR-gate, ↓, is called a Peirce arrow after C. S. Peirce. Charles Sanders Peirce was a theorist of logic, language, communication, and the general theory of signs. He was a great mathematical logician and general mathematician. He was born on September 10, 1839 in Cambridge, MA. Over 57 years his published works consisted of 12, 000 printed pages and 80, 000 known unpublished handwritten pages. His topics had a wide variety ranging from mathematics, to physical science, to economics, to psychology, and different social sciences. HIs father was a math professor at Harvard and contributed greatly to building the math department at that school. Peirce graduated from Harvard in 1859 with a bachelor of science degree in chemistry. For 32 years Peirce was employed by U. S Coast & Geodetic Survey where he mainly surveyed and carried out geodetic investigations. For a period of time Peirce was also working as a logic professor in the math department at JOhn Hopkins University. He was later released from the position due to some issues about his second wife being a gypsy. At an old age Peirce began to study logic more seriously. He first began when he was 12 years of age. He began to think of all issues as problems of logic and went from there. Starting in his freshman year of college when he was 16 years old Peirce began to study philosophy and studied the works of Kant for about three years. He disagreed with Kant and went on to study natural history and philosophy. For the rest of his life his work on logic was extremely varied. He later became one of the greatest logicians who ever lived.
Bibliography Epp, Susanna S. Discrete Mathematics with Applications. 4. Boston, MA: Brooks/Cole, 2011. 1 -820. Print. Burch, Robert, "Charles Sanders Peirce", The Stanford Encyclopedia of Philosophy (Summer 2013 Edition), Edward N. Zalta (ed. ), URL = <http: //plato. stanford. edu/archives/sum 2013/entries/peirce/>.