2nd half of class, MAT 227 - Final

Ace your homework & exams now with Quizwiz!

Suppose you have an odd number of pennies. Prove that you must have an odd number of at least one of the other types of coins.

Suppose you have 22 coins, including 2k nickels, 2j dimes, and 2l quarters (so an even number of each of these three types of coins). The number of pennies you have will then be.22−2k−2j−2l=2(11−k−j−l). But this says that the number of pennies is also even (it is 2 times an integer). Thus we have established the contrapositive of the statement, "If you have an odd number of pennies then you have an odd number of at least one other coin type."

Suppose you break your piggy bank and scoop up a handful of 22 coins (pennies, nickels, dimes and quarters).Prove that you must have at least 6 coins of a single denomination.

Suppose you only had 5 coins of each denomination. This means you have 5 pennies, 5 nickels, 5 dimes and 5 quarters. This is a total of 20 coins. But you have more than 20 coins, so you must have more than 5 of at least one type.

Your "friend" has shown you a "proof" he wrote to show that .1=3. Here is the proof:Proof. I claim that .1=3. Of course we can do anything to one side of an equation as long as we also do it to the other side. So subtract 2 from both sides. This gives .−1=1. Now square both sides, to get .1=1. And we all agree this is true.What is going on here? Is your friend's argument valid? Is the argument a proof of the claim ?1=3? Carefully explain using what we know about logic.

The argument contains a logical fallacy. The issue arises when squaring both sides of the equation (-1=1).While it's true that if a=b , then a2=b2 , the reverse is not necessarily true.

P-->Q is false

The proposition that is false when P is true and q is false otherwise

Simplify the statements below to the point that negation symbols occuronly directly next to predicates.(a) ¬∀x∀y(x < y ∨ y < x).(b) ¬(∃xP(x) → ∀yP(y)).

The required negations are :a) ¬[¬∀x∀y(x<y∨y<x)]≡∀x∃y(x≥y)∧(y≥x)).b) ¬[¬∃xP(x)∀ P(y)) ≡∃xP(x) ∀yP(y)

chromatic number

The smallest number of colors needed for an edge coloring of a graph

The graph G has 7 vertices with degrees 2,2,2,3,4,5,6. How many edges does G have? Could G be planar? If so, how many faces would it have. If not, explain.

The sum of the degrees of all vertices in an undirected graph is equal to twice the number of edges. Therefore, the sum of the degrees in G is 2+2+2+3+4+5+6=24, and the number of edges in G is 24/2=12. Step 2 - G is planar Step 3 - 7 faces

For each sentence below, determine if it's an atomic statement, Molecular statement or statement at all. c. The customers wore shoes and they were socks.

This is a molecular statement. It combines two separate ideas in one sentence. It is composed of multiple atomic statements joined by the conjunction, "and".

Suppose P and Q are the statements: P: Jack passed math. Q: Jill passed math. a. Translate "Jack and Jill both passed math" into symbols. b. Translate "If Jack passed math, then Jill did not" into symbols. c. Translate "P" and "Q" into English. d. Translate "𝀈(P∧Q)--> Q" into English. e. Suppose you know that if Jack passed math, then so did Jill. What can you conclude if you know that: 1. Jill passed math? 2. Jill did not pass math?

a. P∧Q b. P-->𝀈Q c. Jill passed Math or Jack passed Math d. DeMorgan's Law - If Jack and Jill both did not both pass Math, then Jill did e. 1. Jack passed Math 2. Jack did not pass Math either.

Let P(x) be the predicate, "7(x) +1 is even a. Is P(5) true or false? b. What, if anything, can you conclude about x Ȝ P(x) from the truth value of P(5)? c. What if anything can you conclude about x V P(x) from the truth value of P(5)?

a. True b. ȜxP(x) must be true c. VP(x) must be false.

Simplifying negations will be especially useful in the next sectionwhen we try to prove a statement by considering what would happenif it were false. For each statement below, write the negation of thestatement as simply as possible. Don't just say, "it is false that . . . ". (a) Every number is either even or odd. (b) There is a sequence that is both arithmetic and geometric. (c) For all numbers n, if n is prime, then n + 3 is not prime.

hence negations are as follows: a)" There is a number which is not even and not odd. "b) " Every sequence is neither arithmetic nor geometric. "c) " There is a number n which is prime and n+3 is also prime "

p->q is read as:

if p then q implication or conditional, hypothesis (antecedent), conclusion (consequent)

Implication of a statement

if p, then q

P∧Q is true

is only true if both P and Q are true

𝀈P

is read as not P and called negation

p ∨ q

is true if at least P or Q is true (logical disjunction)

contrapositive of an original statement

occurs when you switch the hypothesis and the conclusion in a statement, and negate both statements. In this example, when we switch the hypothesis and the conclusion, and negate both, the result is: If it is not a polygon, then it is not a triangle.

The converse of the statement p -> q is

q -> p

What is the largest number of edges possible in a graph with 10 vertices? What is the largest number of edges possible in a bipartite graph with 11 vertices? What is the largest number of edges possible in a tree with 11 vertices?

the largest number of edges possible in a graph with 11 vertices? 55 n(n-1)/2 = 11(11-1)/2 the largest number of edges possible in a bipartite graph with 11 vertices? = ≤¼•11² = 30the largest number of edges possible in a tree with 11 vertices? = n-1 = 11-1 = 10

CONVERSE

the statement formed by exchanging the hypothesis and conclusion of a conditional statement

𝀈P is true

when P is false

In my safe is a sheet of paper with two shapes drawn on it in coloredcrayon. One is a square, and the other is a triangle. Each shape isdrawn in a single color. Suppose you believe me when I tell you that ifthe square is blue, then the triangle is green. What do you therefore knowabout the truth value of the following statements?(a) The square and the triangle are both blue

(a). The square and the triangle are both blue. We cannot conclude the truth value of this statement based on the given information. It is possible for the square to be blue, but we have no information about the color of the triangle, Unknown

Consider the statement, "If you will give me a cow, then I will give youmagic beans." Decide whether each statement below is the converse,the contrapositive, or neither (b) If I will not give you magic beans, then you will not give me acow.

(b) is True .Explanation:We know , statement A if and only if statement B ,This means statement A happened if statement B happened and the Statement B happened if the statement B happened. Contrapositive statement of if p then q is if if ~q then ~p . We know if p then q then inverse of this statement is If not p then not q.

Consider the statement, "If you will give me a cow, then I will give youmagic beans." Decide whether each statement below is the converse,the contrapositive, or neither (a) If you will give me a cow, then I will not give you magic beans.

Assume as You will give me a cow and as I will give you magic beans. Neither

Inverse Statement ~P -> ~Q

If not p, then not q

Is the following a valid deduction rule?

Make a truth table that includes all three statements in the argument: PQRP→Q Q→R P→(Q∧R) T T T T T T T T F T F F T F T F T F T F F F F F F T T T T T F T F T T T F F T T T T F F F T T T Notice that in every row for which both P→Q and P→R is true, so is .P→(Q∧R). Therefore, whenever the premises of the argument are true, so is the conclusion. In other words, the deduction rule is valid.

Write the negation, converse and contrapositive for each of the statements below. If the door is closed, then the light is off.

Negation: The door is open, then the light is on. Converse: If the light is off then the door is closed. Contrapositive: If the light is on then the door is open.

Write the negation, converse and contrapositive for each of the statements below. If the power goes off, then the food will spoil.

Negation: The power goes off and the food does not spoil. Converse: If the food spoils, then the power went off. Contrapositive: If the food does not spoil, then the power did not go off.

For all integers a and ,b, if a⋅b is even, then a and b are even.

Negation: There are integers a and b for which a⋅b is even but a or b is odd.Converse: For all integers a and ,b, if a and b are even then ab is even.Contrapositive: For all integers a and ,b, if a or b is odd, then ab is odd.

For every integer x and every integer y there is an integer n such that if x>0 then nx>y .

Negation: There are integers x and y such that for every integer ,n, x>0 and .nx≤y. Converse: For every integer x and every integer y there is an integer n such that if nx>y then x>0. Contrapositive: For every integer x and every integer y there is an integer n such that if nx≤y thenx≤0 .

For all real numbers x and ,y, if xy=0 then x=0 or y=0 .

Negation: There are real numbers x and y such that xy=0 but x≠0 and .y≠0.Converse: For all real numbers x and ,y, if x=0 or y=0 then xy=0Contrapositive: For all real numbers x and ,y, if x≠0 and y≠0 then xy≠0

For all natural numbers ,n, if n is prime, then n is solitary.

Negation: There is a natural number n which is prime but not solitary. Converse: For all natural numbers ,n, if n is solitary, then n is prime. Contrapositive: For all natural numbers ,n, if n is not solitary then n is not prime.

Write the negation, converse and contrapositive for each of the statements below. ∀x(x<1→x^2<1)

Negation: ∃x(x<1∧x²≥1) Converse: ∀x(x²<1→x<1) Contrapositive: ∀x(x²<1→x<1)

Is it possible for a planar graph to have 6 vertices, 10 edges and 5 faces? Explain.

No. A (connected) planar graph must satisfy Euler's formula: v−e+f=2. Here .v−e+f=6−10+5=1.

Make a truth table for (PVQ)-->(P∧Q)

P Q (PVQ) (P∧Q) (P∨Q)-->(P∧Q) T T T T T T F T F F F T T F F F F F F F

Determine if the following deduction rule is valid: P ∨ Q ¬P ∴ Q

P Q P ∨ Q ¬P T T T F T F T F F T T T F F F T Yes

Complete a truth table for the statement ¬P→(Q∧R).

P Q R Q∧R ∽P ∽P->Q∧R T T T T F T T T F T F F T F T T F F T F F T FF

P∧Q is read

P and Q is a conjunction

P∨Q is read

P or Q is a disjunction it has the same truth value as not P then Q

P <--> Q is true when

P or Q or both are true or are both false

Suppose you know that the statement "if Peter is not tall, then Quincy is fat and Robert is skinny" is false. What, if anything, can you conclude about Peter and Robert if you know that Quincy is indeed fat?

Peter is not tall and Robert is not skinny. You must be in row 6 in the truth table above.

Are the statements P-->(QVR) and (P-->Q)V (P-->R) are logically equivalent?

Step 1 - 1. P→(Q∨R): This statement reads "If P is true, then at least one of Q or R (or both) must be true." Step 2 -When you analyze both statements, you'll find that they convey the same meaning.

Are the statements P→(Q∨R) and (P→Q)∨(P→R) logically equivalent? Explain your answer.

Step 1 - 1. P→(Q∨R): This statement reads "If P is true, then at least one of Q or R (or both) must be true." In other words, when P is true, either Q, R, or both Q and R can be true for the statement to be satisfied.2. (P→Q) ∨ (P→R): This statement consists of two parts connected by a logical OR:- (P→Q): This part reads "If P is true, then Q must be true." It implies that if P is true, Q must also be true.- (P→R): This part reads "If P is true, then R must be true." It implies that if P is true, R must also be true. Step 2 -When you analyze both statements, you'll find that they convey the same meaning. In the first statement, if P is true, at least one of Q or R (or both) must be true. In the second statement, it's essentially saying that either if P is true, then Q must be true, or if P is true, then R must be true.So, if P is true, in both cases, either Q, R, or both Q and R will also be true. And if P is false, then the implications from P to Q or R are trivially satisfied since false implies anything. In both scenarios, the truth values of the statements will be the same.

Prove the statement: For all integers ,n, if 5n is odd, then n is odd. Clearly state the style of proof you are using.

Suppose that 5n is an odd number and n is an even number.n=2k for some k∈Z5n=10k5n=2×5kn=2×m wherem=5k∈ZThus, 5n is also an even number, which is giving a contradiction.Therefore, if 5n is odd, then n is odd.

(d) If the triangle is green, then the square is blue. We cannot conclude the truth value of this statement based on the given information. The initial statement only provides information about the triangle being green if the square is blue, but it doesn't tell us anything about the square being blue if the triangle is green, unknown

Consider the statement, "If you will give me a cow, then I will give youmagic beans." Decide whether each statement below is the converse,the contrapositive, or neither (d) If you will not give me a cow, then I will not give you magicbeans.

(d) neither

Determine if the following is a valid deduction rule: (P ∧ Q) → R ¬P ∨ ¬Q ∴ ¬R

1. (P ∧ Q) → R: Premise2. ∽P ∨ ∽Q: Premise3. ∽(P ∧ Q) : 2, DeMorgans4. ∽R Yes

If 14 people each shake hands with each other, how many handshakes took place?

14 people P₁, P₂, P₃ ...P₁₄ shakes hands with each other then, P₁ has 13 choices P₂ has 12 choices P₁₂ has 2 choices P₁₃ has 1 choice P₁₄ has no choice hence, the total number of handshakes must be 1+2+3+4+...+11+12+13+14 = 105nowin terms of graph theory any two people shakes hands with each other If correspondance to complete K₁₄ No. of handshakes must be equal to the number of edges in K₁₄, which is equal to 14(14-1)/2 = 14*13/2 = 91

vertex coloring

A Vertex coloring of a graph G is an assignment of labels which can be thought of as "colors" to the vertices of G so that vertices joined by an edge get different labels (colors).

Complete bipartite graph

A bipartite graph in which every vertex in one set is joined to every vertex in the other set

complete graph

A graph in which each of the n vertices is connected to every other vertex.

Subgraph

A subgraph is a subset of a graph's edges (and associated vertices) that constitutes a graph.

Tommy Flanagan was telling you what he ate yesterday afternoon. Hetells you, "I had either popcorn or raisins. Also, if I had cucumbersandwiches, then I had soda. But I didn't drink soda or tea." Of courseyou know that Tommy is the worlds worst liar, and everything he saysis false. What did Tommy eat?

Because Tommy always lies, we know that each of these statements is false. The negation of each statement is as follows:Not P: Tommy did not have either popcorn or raisins. This means Tommy had neither popcorn nor raisins.Not (If Q, then R): This is equivalent to Q and Not R. So, Tommy had cucumber sandwiches but did not have soda.Not (Not R and Not T): This is equivalent to R or T. So, Tommy had either soda or tea.Combining these negated statements, we can conclude that:Tommy did not eat popcorn or raisins, he ate cucumber sandwiches, and he had either soda or tea. However, given that he had cucumber sandwiches and did not have soda (from the second negated statement), he must have had tea.So, Tommy ate cucumber sandwiches and drank tea.

For which n≥3 is the graph Cn bipartite?

Bipartite graph A graph is bipartite if the vertices of the graph can be partitioned into two sets X and Y such that every edge of the graph contains one end in X and Y.Cycle graph is the graph represented by which contains a unique cycle in the graph. Explanation:Bipartite graph contains no odd cycles. The graph which contains odd cycles are not bipartite.We can determine a cycle is odd or even by the number of edges in it. Step 2Since the bipartite graph contains no odd cycles we get, the graph which are containing odd cycles is not bipartite. The cycle graph contains odd and even number of vertices and cycles. For odd number of vertices in the cycle graph it contains odd cycles, so the graph become not bipartite. So here we get for even numbers the graph contains no odd cycles, so the graph is bipartite. So, the graph is bipartite if the n is even. So, we get. is bipartite for all even values of n.

Bipartite graph

Consists of two sets of vertices X and Y. The edges only join vertices in X to vertices in Y, not vertices within a set.

You come across four trolls playing bridge. They declare: Troll 1: All trolls here see at least one knave. Troll 2: I see at least one troll that sees only knaves. Troll 3: Some trolls are scared of goats.Troll 4: All trolls are scared of goats.Are there any trolls that are not scared of goats? Recall, of course, that all trolls are either knights (who always tell the truth) or knaves (who always lie).

Troll 1: All trolls here see at least one knaveThis cannot be false. If it was, then someone would see only knights. But in this case, both Troll 2 and Troll 1 would be knaves. Thus there are at least two knaves. There are two knights and two knaves.Troll 2: I see at least one troll that sees only knaves.This cannot be true. If it was, then everyone this supposed troll sees would be a knave, including Troll 2. So Troll 2 is a knave and his statement is false. Everyone he sees must see at least one knight. In particular, there must be at least two knights in the group.Troll 3: Some trolls are scared of goats;Troll 4: All trolls are scared of goats.So If Troll 4 is a knight, so is Troll 3.Thus some trolls are scared of goats but not all trolls, so some are not scared of goats.

Suppose P(x) is some predicate for which the statement ∀xP(x) is true.Is it also the case that ∃xP(x) is true? In other words, is the statement∀xP(x) → ∃xP(x) always true? Is the converse always true? Assumethe domain of discourse is non-empty.

True

Is it possible for two different (non-isomorphic) graphs to have the same number of vertices and the same number of edges? What if the degrees of the vertices in the two graphs are the same (so both graphs have vertices with degrees 1, 2, 2, 3, and 4, for example)? Draw two such graphs or explain why not.

Yes, it is possible for both.Yes, if the degrees of the vertices are also same then they can't be non-isomorphic as they become isomorphicHence, no two graphs that are non-isomorphic can have the same number of edges, vertices of degrees of vertices all at once.

How many coins would you need to scoop up to be sure that you either had 4 coins that were all the same or 4 coins that were all different? Prove your answer.

You need 10 coins. You could have 3 pennies, 3 nickels, and 3 dimes. The 10th coin must either be a quarter, giving you 4 coins that are all different, or else a 4th penny, nickel or dime. To prove this, assume you don't have 4 coins that are all the same or all different. In particular, this says that you only have 3 coin types, and each of those types can only contain 3 coins, for a total of 9 coins, which is less than 10.

Euler Circuit

a circuit that travels along each edge of a graph once and only once

Euler Path

a path that travels along each edge of a graph once and only once

statement

a sentence that is either true or false

atomic statement

a type of declarative sentence which is either true or false -- cannot be broken down into simpler sentences

Again, suppose the statement "if the square is blue, then the triangle is green" is true. This time however, assume the converse is false. Classify each statement below as true or false (if possible). (a) The square is blue if and only if the triangle is green.

a) First we assume that ,The square is blue .Then it is given that , The triangle is greenBut If the triangle is green does not imply the square is blue. Hence , the statement (a) is false

Consider the following two attached graphs: a) Does f define an isomorphism between Graph 1 and Graph 2? b) Define a new function g (with g ≠ f) that defines an isomorphism between Graph 1 andGraph 2. c) Is the graph pictured below in Figure 1 isomorphic to Graph 1 and Graph 2? Explain

a) Therefore the degree of vertices in both graphs are given belowIn a graph G1, we havedeg(a)=2,deg(b)=5,deg(c)=2,deg(d)=3,deg(e)=3,deg(f)=3,deg(g)=2In a graph G2, we havedeg(v1)=3,deg(v2)=2,deg(v3)=3,deg(v4)=2,deg(v5)=5,deg(v6)=2,deg(v7)=3Now as given degree of vertex deg(c)=2in G1 but deg(v1)=3 in G2So f(c)≠v1Hence f is not an isomorphism (b) Yes, there exists an isomorphism g≠f defined as belowg(a)=v2,g(b)=v5,g(c)=v4,g(d)=v1,g(e)=v3,g(f)=v7,g(g)=v6( c) As in given graph shown in figure 1, there are only two vertices of degree two(2), but in G1andG2, there are three three vertices of degree two(2). Hence by definition, this is not isomorphic to neither G1 nor G2.

Consider the statement: for all integers ,n, if n is odd and n≤8 then n is negative or n∈ {1,3,5,7}. a. Is the statement true? Explain why.

a) This statement is ture. reason: since n is odd and m <= 8, so n must be negative odd number and {1,3,5,7}

Determine whether each molecular statement below is true or false, orwhether it is impossible to determine. Assume you do not know what my favorite number is (but you do know that 13 is prime). (a) If 13 is prime, then 13 is my favorite number. (b) If 13 is my favorite number, then 13 is prime. (c) If 13 is not prime, then 13 is my favorite number. (d) 13 is my favorite number or 13 is prime. (e) 13 is my favorite number and 13 is prime. (f) 7 is my favorite number and 13 is not prime. (g) 13 is my favorite number or 13 is not my favorite number.

a) not enough information, b) True, c) True, d) not enough information, e) not enough information, f) False, g) True

Simplify the following: a. ¬(¬(P∧¬Q)→¬(¬R∨¬(P→R))).b. ¬∃x¬∀y¬∃z(z=x+y→∃w(x-y=w))

a. (¬P V¬Q)∧(¬R∨(P∧¬R)) b. ∀x∀y.∀z(z=x+y ∧ ∀w (x-y≠w))

Translate into English: (a) ∀x(E(x) → E(x + 2)). (b) ∀x∃y(sin(x) =y). (c) ∀y∃x(sin(x) = y). (d) ∀x∀y(x^3 = y^3 → x = y)

a. Any even number plus two is an even number. b. For every number x, there exists y that sin(x) = y holds c. For every numbery, there exists x that sin(x) = y holds d. For every x and y, if x^3 = y^3 then x=y holds

Which of the following statements are equivalent to the implication,"if you win the lottery, then you will be rich," and which are equivalentto the converse of the implication? (a) Either you win the lottery or else you are not rich. (b) Either you don't win the lottery or else you are rich. (c) You will win the lottery and be rich. (d) You will be rich if you win the lottery. e) You will win the lottery if you are rich. (f) It is necessary for you to win the lottery to be rich. (g) It is sufficient to win the lottery to be rich. (h) You will be rich only if you win the lottery. (i) Unless you win the lottery, you won't be rich. (j) If you are rich, you must have won the lottery. (k) If you are not rich, then you did not win the lottery. (l) You will win the lottery if and only if you are rich

a. Converseb. Implicationc. Neither implication not conversed. Implicatione. Conversef. Converseg. Implicationh. Conversei. Conversej. Conversek. Implicationl. Neither

Consider the statement: for all integers ,n, if n is even then 8n is even. a. Prove the statement. What sort of proof are you using? b. Is the converse true? Prove or disprove.

a. Direct proof.Proof. Let n be an integer. Assume n is even. Then n=2k for some integer k. Thus 8n = 16k = 2(8k). Therefore 8n is even. b. The converse is false. That is, there is an integer n such that 8n is even but n is odd. For example, consider n=3. Then 8n=24 which is even but n=3 is odd.

Consider the statement "for all integers a and b, if a+b is even, then a and b are even" a. Write the contrapositive of the statement. b. Write the converse of the statement. c. Write the negation of the statement. d. Is the original statement true or false? Prove your answer. e. Is the contrapositive of the original statement true or false? Prove your answer.f. Is the converse of the original statement true or false? Prove your answer. g. Is the negation of the original statement true or false? Prove your answer.

a. For all integers a and ,b, if a or b is not even, then a+b is not even. b. For all integers a and ,b, if a and b are even, then a+b is even. c. There are numbers a and b such that a+b is even but a and b are not both even. d. False. For example, a=3 and .b=5. ,a+b=8, but neither a nor b are even. e. False, since it is equivalent to the original statement. f. True. Let a and b be integers. Assume both are even. Then a=2k and b=2j for some integers k and .j. But then a+b=2k+2j=2(k+j) which is even. g. True, since the statement is false.

Consider the statement "If Charles eats pizza, then he drinksCoke."(a) Write the converse of the statement.(b) Write an inverse to the original the statement.(c) Which of the following is an inverse to the original statement?

a. If Charles is drinking Coke, then he is eating pizza.b. If Charles is not eating pizza, then he is not drinking Coke.c. If Charles is not drinking Coke, then he is not eating pizza.

For each sentence below, determine if it's an atomic statement, Molecular statement or statement at all.a. Customers must wear shoes.

a. Not a statement

a. Not a statement at all. It is a phrase, not expressing a complete thought or making a claim.

Classify each of the sentences below as an atomic statement, a molecular statement, or a statement at all. If the statement is molecular state what kind it is (conjunction, disjunction, conditional, biconditional, negation) a. The sum of the first 100 odd positive integers. b. Everybody needs somebody sometimes. c. The Broncos will win the Super Bowl or I'll eat my hat. d. We can have donuts for dinner, but only if it rains. e. Every natural number greater than 1 is either prime or composite. f. This sentence is false.

a. Not a statement at all. It is a phrase, not expressing a complete thought or making a claim. b. Not a statement at all. It is a general expression or phrase, not expressing a complete thought or making a claim. c. Molecular statement. It is a disjunction. (specifically an inclusive disjunction). It combines two statements: "The Broncos will win the Super Bowl" and "I'll eat my hat." The statement expresses that at least one of these statements is true. d. Molecular statement. It is a bi - conditional statement. It combines two statements. "We have donuts for dinner" and "It rains." The statement expresses a condition (if it rains) under which the first statement (having donuts for dinner) is true. e. Atomic Statement f. Not a statement at all. It is a self-referential paradoxical statement, known as the liar paradox. It leads to the contradiction and does not express a well-formed claim.

Let P(x) be the predicate, "4(x) + 1 is even." (a) Is P(5) true or false? (b) What, if anything, can you conclude about ∃xP(x) from the truthvalue of P(5)? (c) What, if anything, can you conclude about ∀xP(x) from the truthvalue of P(5)?

a. P(5) = 4x(5)+1 = 21P(5) is false. b. He cannot conclude anything about the truth value from ∃xP(x) of P(5) c. If we observe for any value of x either odd or even, P(x) is false

Let P(x) be the predicate, "4x + 1 is even."(a) Is P(5) true or false?(b) What, if anything, can you conclude about ∃xP(x) from the truthvalue of P(5)?(c) What, if anything, can you conclude about ∀xP(x) from the truthvalue of P(5)?

a. P(5) = 4x(5)+1 = 21P(5) is false. b. He cannot conclude anything about the truth value from ∃xP(x) of P(5) c. If we observe for any value of x either odd or even, P(x) is false

For each of the statements below, say what method of proof you should use to prove them. Then say how the proof starts and how it ends. Bonus points for filling in the middle. a. There are no integers x and y such that x is a prime greater than 5 and x=6y+3. b. For all integers ,n, if n is a multiple of 3, then n can be written as the sum of consecutive integers. c. For all integers a and ,b, if a2+b2 is odd, then a or b is odd.

a. Proof by contradiction. Start of proof: Assume, for the sake of contradiction, that there are integers x and y such that x is a prime greater than 5 and x=6y+3. End of proof: ... this is a contradiction, so there are no such integers. b. Direct proof. Start of proof: Let n be an integer. Assume n is a multiple of 3. End of proof: Therefore n can be written as the sum of consecutive integers. c. Proof by contrapositive. Start of proof: Let a and b be integers. Assume that a and b are even. End of proof: Therefore a2+b2 is even.

Suppose P(x, y) is some binary predicate defined on a very smalldomain of discourse: just the integers 1, 2, 3, and 4. For each of the 16pairs of these numbers, P(x, y) is either true or false, according to thefollowing table (x values are rows, y values are columns). For example, P(1, 3) is false, as indicated by the F in the first row,third column.Use the table to decide whether the following statements are trueor false. (a) ∀x∃yP(x, y). Introduction and Preliminaries (b) ∀y∃xP(x, y). (c) ∃x∀yP(x, y). (d) ∃y∀xP(x, y)

a. True - Now from the table, x=3 and for row 3 if it is taken, y=1,2,3,4 then P(3,1), P(3,2), P(3,3) and P(3,4) are all trueb. True - (2,2)c. False - (2,3) is trued. True - P(4,y) is true

For each sentence below, determine if it's an atomic statement, Molecular statement or statement at all. a. Customers must wear shoes. b. The customers wore shoes. c. The customers wore shoes and they were socks.

a. not a statement at all b. This is an atomic statement. It presents a single, complete idea. c. This is a molecular statement. It combines two separate ideas in one sentence. It is composed of multiple atomic statements joined by the conjunction, "and".

For a given predicate P(x), you might believe that the statements∀xP(x) or ∃xP(x) are either true or false. How would you decide ifyou were correct in each case? You have four choices: you could givean example of an element n in the domain for which P(n) is true orfor which P(n) if false, or you could argue that no matter what n is,P(n) is true or is false. (a) What would you need to do to prove ∀xP(x) is true?

a. show that for every element n in the domain that P(n) is true (iii)

a. show that for every element n in the domain that P(n) is true (iii) b. Give an example of an element n in the domain for which P(n) is false (ii) c. Take an element from the domain for which P(n) is true. (i) d. Show that for every element n in the domain that P(n) is false. (iv)

We can also simplify statements in predicate logic using our rules for passing negations over quantifiers, and then applying propositional logical equivalence to the "inside" propositional part. Simplify the statements below (so negation appears only directly next to predicates). (a) ¬∃x∀y(¬O(x) ∨ E(y)). (b) ¬∀x¬∀y¬(x < y ∧ ∃z(x < z ∨ y < z)). (c) There is a number n for which no other number is either less n than or equal to n. (d) It is false that for every number n there are two other numbers which n is between.

a. ∀x∃y(O(x) ∧¬ E(y)) b. ∃x∀y(x≥ y∨∀z (x≥z∧y≥z) c. ∃n∀m(m>n) d. ∃n∀m∀k(m≤nVn≤k)

P <-->Q is read

as P iff Q and called biconditional

Again, suppose the statement "if the square is blue, then the triangle is green" is true. This time however, assume the converse is false. Classify each statement below as true or false (if possible). (b) The square is blue if and only if the triangle is not green.

b) Given ,The square is blue if and only if the triangle is not green .This means , if square is blue then the triangle is not green this statement is true as ,As the converse of the statement "The square is blue , then the triangle is green "Is not true , that if the triangle is green then the square is not blue .Then by negation of this statement is ,If the square is blue then the triangle is not green. Now , If the triangle is not green then the square is blue, this is true statement , as in the statement if the square is blue then the triangle is green, the converse is not true .That is if the triangle is green then the square is not blue. Then by the inverse of the above statement is, if the triangle is not green then the square is blue . Hence the statement (b) is True .Explanation:We know , statement A if and only if statement B ,This means statement A happened if statement B happened and the Statement B happened if the statement B happened. Contrapositive statement of if p then q is if if ~q then ~p . We know if p then q then inverse of this statement is If not p then not q.

Consider the statement: for all integers ,n, if n is odd and n≤8 then n is negative or n∈ {1,3,5,7}. b. Write the negation of the statement. Is it true? Explain.

b) the negation is:n is integer, n is odd and n <= 8, then n is positive or n is not {1,3,5,7} th this negation of the statement is false. {1,3,5,7} is <=8.

For a given predicate P(x), you might believe that the statements∀xP(x) or ∃xP(x) are either true or false. How would you decide ifyou were correct in each case? You have four choices: you could givean example of an element n in the domain for which P(n) is true orfor which P(n) if false, or you could argue that no matter what n is,P(n) is true or is false. b) What would you need to do to prove ∀xP(x) is false?

b. Give an example of an element n in the domain for which P(n) is false (ii)

For each sentence below, determine if it's an atomic statement, Molecular statement or statement at all..b. The customers wore shoes.

b. This is an atomic statement. It presents a single, complete idea.

b. atomic statement

c) Given statement "if the square is blue ,then the triangle is green "But converse of the given statement is not true .That is if the triangle is green then the square is not blue .So , The square is blue is not always true .Hence , the statement (c) is false

Consider the statement: for all integers ,n, if n is odd and n≤8 then n is negative or n∈ {1,3,5,7}. State the contrapositive of the statement. Is it true? Explain.

c)The contrapositive statement is : if n is positive, or n is not {1,3,5,7} then for all integers, n is even and n>8. this contrapositive statement is false. The reason is even number is not 1,3,5,7. since n>8, n can't be 1,3,5,7 and n is positive. It satisfied the contrapositive statement.

For a given predicate P(x), you might believe that the statements∀xP(x) or ∃xP(x) are either true or false. How would you decide ifyou were correct in each case? You have four choices: you could givean example of an element n in the domain for which P(n) is true orfor which P(n) if false, or you could argue that no matter what n is,P(n) is true or is false. (c) What would you need to do to prove ∃xP(x) is true?

c. Take an element from the domain for which P(n) is true. (i)

Molecular Statement

cannot be divided into smaller statements...

Among a group of 7 people, is it possible for everyone to be friends with exactly two of the people of the group?

considering people as vertices v₁, v₂, ...v₇ and edges between them as friendship is v₁ and v₂ are friends, yes it is possible for two people.Now each person has 3 friendsbut by the handshaking theorem, the total degrees of the graph must be even. Thus such a graph isn't possible.This is not possible in a group of 7 people to each have 3 friends.

d) Given that , if the square is blue then the triangle is green. As the converse is not true, so If the triangle is green then the square is not blue. For all cases , the triangle is green. Hence , the statement (d) is true

Consider the statement: for all integers ,n, if n is odd and n≤8 then n is negative or n∈ {1,3,5,7}. State the converse of the statement. Is it true? Explain.

d) the converse of the statement is : for all integer n,if n is negative or n is {1,3,5,7},then n is odd and n is <= 8. the converse statement is false. - 2 is negative, but n is even. so the converse statement is false

d. molecular - biconditional

e. Atomic statement

f. Not a statement at all. It is a self-referential paradoxical statement, known as the liar paradox. It leads to the contradiction and does not express a well-formed claim.

See all study sets

2nd half of class, MAT 227 - Final

Related study sets

Chapter 26

Physics 2 Exam 3 - Quizzes - Ch 16, 17, & 18

MedSurg 4

Sentence Grammar Diagnostic Post-Test English 102 Wes Mantooth LWIT LWTCH

Ancient Greece

Previously Missed Q's 7

smell (olfactory)

second study

Respiratory nclex questions

Patho Hesi

Research, Chapter 12, Populations and Samples in Research

Biology 111 Chapter 9

ECON - Aggregate Demand Smartbook

CHAPTER 5: RULES AND REGULATIONS

MLT ASCP Practice Questions: Urinalysis & Body Fluids

Chapter 21: Family-Centered Care of the Child During Illness and Hospitalization

Introduction to Epidemiology

Pain questions

Econ Exam 2

EXAM 1