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nature of mathematics

Questions and Answers of Nature Of Mathematics

Show that each set in Problems 45–48 is infinite by placing it into a one-to-one correspondence with a proper subset of itself. W
{285, 280, 275, . . .}Show that each set in Problems 35–44 has cardinality
{1, 3, 9, 27, 81, . . .}Show that each set in Problems 35–44 has cardinality
Set of positive multiples of 5Show that each set in Problems 35–44 has cardinality
Set of odd counting numbersShow that each set in Problems 35–44 has cardinality
Show that each set in Problems 35–44 has cardinality Z
Show that each set in Problems 35–44 has cardinality 80.
{y | y ∈ Z and is divisible by 2}Show that each set in Problems 35–44 has cardinality
Show that each set in Problems 35–44 has cardinality {XEN}
a. If A represents the letters in the English alphabet, what is the cardinality of A?b. If B represents the number of U.S. states, what is the cardinality of B?c. What the cardinality of the set
Give the cardinality of each set in Problems 11–20. [1 1 1 1 2 3 4 5 }
Human blood is typed Rh+ (positive blood) or Rh- (negative blood). This Rh factor is called an antigen. There are two other antigens known as A and B types. Blood lacking both A and B types is called
Using the Venn diagram in Figure 2.17, specify which region is described by Figure 2.17 ANBUCUDUE.
Using the Venn diagram in Figure 2.17, specify which region is described by Figure 2.17 AUBUCUDUE.
On the NBC Nightly News on Thursday, May 25, 1995, Tom Brokaw read a brief report on computer use in the United States. The story compared computer users by ethnic background, and Brokaw reported
Matt E. Matic was applying for a job. To determine whether he could handle the job, the personnel manager sent him out to poll 100 people about their favorite types of TV shows. His data were as
In a recent survey of 100 persons, the following information was gathered:59 use shampoo A.51 use shampoo B.35 use shampoo C.24 use shampoos A and B.19 use shampoos A and C.13 use shampoos B and C.11
In Problems 45–50, use Venn diagrams to prove or disprove each statement. Remember to draw a diagram for the left side of the equation and another for the right side. If the final shaded portions
In Problems 45–50, use Venn diagrams to prove or disprove each statement. Remember to draw a diagram for the left side of the equation and another for the right side. If the final shaded portions
In Problems 45–50, use Venn diagrams to prove or disprove each statement. Remember to draw a diagram for the left side of the equation and another for the right side. If the final shaded portions
In Problems 45–50, use Venn diagrams to prove or disprove each statement. Remember to draw a diagram for the left side of the equation and another for the right side. If the final shaded portions
In Problems 39–44, use set notation to identify the shaded region. U A C B
In Problems 39–44, use set notation to identify the shaded region. U A C B
In Problems 39–44, use set notation to identify the shaded region. U A C B
In Problems 39–44, use set notation to identify the shaded region. U A C
In Problems 39–44, use set notation to identify the shaded region. U A B
In Problems 39–44, use set notation to identify the shaded region. U A
Draw a Venn diagram showing people in a classroom wearing some black, people wearing some blue, and people wearing some brown.
Consider the sets X and Y. Write each of the statements in Problems 11–18 in symbols.An intersection of complements
Decide whether each statement in Problems 46–54 is true or false. Give reasons for your answers.a. 1 ∈ {1, 2, 3, 4, 5}b. {1} ∈ {1, 2, 3, 4, 5}
Decide whether each statement in Problems 46–54 is true or false. Give reasons for your answers.a. {math, history} ⊂ {high school subjects}b. { } ⊆ {Jeff, Maureen, Terry}
Decide whether each statement in Problems 46–54 is true or false. Give reasons for your answers.a. White ∈ {colors of the rainbow}b. White ∈ {colors of the U.S. flag}
Decide whether each statement in Problems 46–54 is true or false. Give reasons for your answers.a. {m, a, t, h} ⊆ {h, t, a, m}b. {m, a, t, h} ⊂ {h, t, a, m}
Build a simple device that will add single-digit numbers.
Between now and the end of the course, look for logical arguments in newspapers, periodicals, and books. Translate these arguments into symbolic form. Turn in as many of them as you find. Be sure to
All squares are rectangles.All rectangles are quadrilaterals.All quadrilaterals are polygons.In Problems 17–19, form a valid conclusion using all the premises given in each problem.
Let p: p is a prime number; q: P + 2 is a prime number.Translate the following statements into verbal form.a. p → qb. (pVq) ^ [~(p ^ q)]
(p ∧ q) ∨ (p ∧ r)Using AND-gates, OR-gates, and NOT-gates, design circuits that would find the truth values for the statements in Problems 32–37.
(p ∧ q) ∨ rUsing AND-gates, OR-gates, and NOT-gates, design circuits that would find the truth values for the statements in Problems 32–37.
q → ∼pFirst, build a truth table and then, using switches, design a circuit that would find the truth values for the statements in Problems 26–31 (answers are not unique).
∼(p ∧ q)Using both switches and simplified gates, design a circuit that would find the truth values for the statements in Problems 20–25 (answers are not unique).
p ∧ qUsing both switches and simplified gates, design a circuit that would find the truth values for the statements in Problems 20–25 (answers are not unique).
Alice: “I don’t know.”Ben: “I don’t win.”Cole: “I tie as a winner.”In Problems 31–36, state what you can infer about the cards from the given statements.
Alice: “I don’t know.”Ben: “I don’t win.”Cole: “I win.”In Problems 31–36, state what you can infer about the cards from the given statements.
Alice: “I don’t know.”Ben: “I lose.”Cole: “I lose.”In Problems 31–36, state what you can infer about the cards from the given statements.
I visited a beautiful flower garden yesterday and counted exactly 50 flowers. Each flower was either red or yellow, and the flowers were not all the same color. My friend made the following
A group of 50 teachers and school administrators attended a convention. If any two persons had been picked at random, at least one of the two would be a teacher. From this information, is it possible
A fox, hunting for a morsel of food, spotted a huge bear about 100 yards due east of him. Before the hunter could become the hunted, the crafty fox ran due north for 100 yards but then realized the
Three people—call them Al, Bob, and Cary—were standing together at a bus stop. A stranger asked Al, “Are you a knight or a knave?” Al answered, but the stranger could not make out what he
Given: MIIIUUIIIII; Prove: MIIUUse the definitions and postulates given in Example 2 to prove the theorems in Problems 9–14. Give both statements and reasons.
Given: MI; Prove: MUIUse the definitions and postulates given in Example 2 to prove the theorems in Problems 9–14. Give both statements and reasons.
Prove by constructing a truth table. What is the name we give to this type of reasoning? [(pq) Ap]q P.
(Let x be an integer and y a nonzero integer for this argument.) If Q is a rational number, then Q = x/y , where x/y is a reduced fraction.Q ≠ x/y , where x/y is a reduced fraction.Therefore, Q is
No students are enthusiastic.You are enthusiastic.Therefore, you are not a student.Use a symbolic argument in Problems 29–32 to determine if the argument is valid or invalid.
All mathematicians are eccentrics.All eccentrics are rich.Therefore, all mathematicians are rich.Use a symbolic argument in Problems 29–32 to determine if the argument is valid or invalid.
If Al believed that cars would ruin the planet, then he wouldnot travel by car.Al travels by car.Therefore, Al does not believe that cars will ruin the planet.Determine whether each argument in
If you like beer, you’ll like Bud.You don’t like Bud.Therefore, you don’t like beer.Determine whether each argument in Problems 9–28 is valid or invalid. If valid, name the type of reasoning
If you get a fill-up of gas, you will get a free car wash.You get a fill-up of gas.Therefore, you will get a free car wash.Determine whether each argument in Problems 9–28 is valid or invalid. If
If Ron uses Slippery oil, then his car is in good running condition.Ron’s car is in good running condition.Therefore, Ron uses Slippery oil.Determine whether each argument in Problems 9–28 is
If Todd eats Krinkles cereal, then he has “extra energy.”Todd has “extra energy.”Therefore, Todd eats Krinkles cereal.Determine whether each argument in Problems 9–28 is valid or invalid.
If 2x – 4 = 0, then x = 2.x ≠ 2Therefore, 2x – 4 ≠ 0.Determine whether each argument in Problems 9–28 is valid or invalid. If valid, name the type of reasoning and if invalid, determine the
If Mary does not have a little lamb, then she has a big bear.Mary does not have a big bear.Therefore, Mary has a little lamb.Determine whether each argument in Problems 9–28 is valid or invalid. If
If Al understands a problem, it is easy.This problem is not easy.Therefore, Al does not understand this problem.Determine whether each argument in Problems 9–28 is valid or invalid. If valid, name
If Al understands logic, then he enjoys this sort of problem.Al does not understand logic.Therefore, Al does not enjoy this sort of problem.Determine whether each argument in Problems 9–28 is valid
Blue-chip stocks are safe investments.Stocks that pay a high rate of interest are safe investments.Therefore, blue-chip stocks pay a high rate of interest.Determine whether each argument in Problems
If Fermat’s last theorem is ever proved, then my life is complete.Fermat’s last theorem was proved in 1994.Therefore, my life is complete.Determine whether each argument in Problems 9–28 is
All snarks are fribbles.All fribbles are ugly.Therefore, all snarks are ugly.Determine whether each argument in Problems 9–28 is valid or invalid. If valid, name the type of reasoning and if
If I inherit $1,000, I will buy you a cookie.I inherit $1,000.Therefore, I will buy you a cookie.Determine whether each argument in Problems 9–28 is valid or invalid. If valid, name the type of
Test the validity of the following argument.If a person goes to college, he will make a lot of money.You do not go to college.Therefore, you will not make a lot of money.
The money is available or I will not take my vacation.Use the tautology (p → q) ⇔ (∼p ∨ q) to write each statement in Problems 29–34 in an equivalent form.
The sun is shining or I will not go to the park.Use the tautology (p → q) ⇔ (∼p ∨ q) to write each statement in Problems 29–34 in an equivalent form.
If the cherries have turned red, then they are ready to be picked.Use the tautology (p → q) ⇔ (∼p ∨ q) to write each statement in Problems 29–34 in an equivalent form.
If I go, then I paid $100.Use the tautology (p → q) ⇔ (∼p ∨ q) to write each statement in Problems 29–34 in an equivalent form.
If by the mere force of numbers a majority should deprive a minority of any clearly written constitutional right, it might, in a moral point of view, justify revolution. (Abraham Lincoln)Translate
No person who has once heartily and wholly laughed can be altogether irreclaimably bad. (Thomas Carlyle)Translate the statements in Problems 19–28 into symbols. For each simple statement, indicate
Be nice to people on your way up ’cause you’ll meet ’em on your way down. (Jimmy Durante)Translate the statements in Problems 19–28 into symbols. For each simple statement, indicate the
Either I will invest my money in stocks or I will put it in a savings account.Translate the statements in Problems 19–28 into symbols. For each simple statement, indicate the meanings of the
No man is an island.Translate the statements in Problems 19–28 into symbols. For each simple statement, indicate the meanings of the symbols you use.
I cannot go with you because I have a previous engagement.Translate the statements in Problems 19–28 into symbols. For each simple statement, indicate the meanings of the symbols you use.
To obtain the loan, I must have an income of $85,000 per year.Translate the statements in Problems 19–28 into symbols. For each simple statement, indicate the meanings of the symbols you use.
I will not buy a new house unless all provisions of the sale are clearly understood.Translate the statements in Problems 19–28 into symbols. For each simple statement, indicate the meanings of the
Neither smoking nor drinking is good for your health.Translate the statements in Problems 19–28 into symbols. For each simple statement, indicate the meanings of the symbols you use.
No p is qVerify the indicated definition in Problems 15–18 from Table 3.9 using a truth table.Table 3.9 Either p or q pq (pvq)^~(p^q) TT F TF FT FF T T F Neither p nor q p unless q ~q p ~(pvq) F T
(p → q) ↔ (~ p ∨ q)Use truth tables in Problems 7–14 to determine whether the given compound statement is a tautology.
All good people go to heaven.Translate the sentences in Problems 31–38 into if-then form.
All prime numbers greater than 2 are odd numbers.Translate the sentences in Problems 31–38 into if-then form.
All triangles are polygons.Translate the sentences in Problems 31–38 into if-then form.
If you brush your teeth with Smiles toothpaste, then you will have fewer cavities.Write the converse, inverse, and contrapositive of the statements in Problems 27–30.
I will go Saturday if I get paid.Write the converse, inverse, and contrapositive of the statements in Problems 27–30.
If you break the law, then you will go to jail.Write the converse, inverse, and contrapositive of the statements in Problems 27–30.
a. ∼p → ∼qb. ∼t → sWrite the converse, inverse, and contrapositive of the statements in Problems 27–30.
Repeat Example 6, except this time assume that you do not complete the worksheet, but all other statements are true.Data from Example 6The following sentence is found on a tax form:If you do not
A copy of The Bootleg Adjunct Newsletter of the Faculty Association at Santa Rosa Junior College said, “As you well know, next fall’s class schedule has been reduced –10%.” What do you think
If the sign on the left means no parking, what do you think the sign on the right means? R
(p ∧ q) ∧ ∼rConstruct a truth table for the statements given in Problems 5–22.
(p ∨ q) ∨ rConstruct a truth table for the statements given in Problems 5–22.
(p ∧ q) → pConstruct a truth table for the statements given in Problems 5–22.
P ∨ (p→q)Construct a truth table for the statements given in Problems 5–22.
(p ∧ ~q) ∧ pConstruct a truth table for the statements given in Problems 5–22.
r ∧ s ∨ ∼sConstruct a truth table for the statements given in Problems 5–22.

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