A Transition to Advanced Mathematics
A Transition to Advanced Mathematics
8th Edition
ISBN: 9781285463261
Author: Douglas Smith, Maurice Eggen, Richard St. Andre
Publisher: Cengage Learning
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Textbook Question
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Chapter 1.1, Problem 3E

Make a truth table for each of the following propositional forms.

  1. P ~ P
  2. P ~ P
  3. P ~ Q
  4. P ( Q ~ Q )
  5. ( P P ) ~ Q
  6. ~ ( P Q )
  7. ( P ~ Q ) R
  8. ~ P ~ Q
  9. P ( Q R )
  10. ( P Q ) ( P R )
  11. P P
  12. ( P Q ) ( R ~ S )

(a)

Expert Solution
Check Mark
To determine

To make: truth table for PP .

Answer to Problem 3E

    PQPQPP
    TTFFF
    TFFTF
    FTTFF
    FFTTF

Explanation of Solution

Given information: PP

Calculation:

The required truth table for PP is follows:

    PQPQPP
    TTFFF
    TFFTF
    FTTFF
    FFTTF

(b)

Expert Solution
Check Mark
To determine

To make: truth table for PP .

Answer to Problem 3E

    PQPQPP
    TTFFT
    TFFTT
    FTTFT
    FFTTT

Explanation of Solution

Given information: PP

Calculation:

The required truth table for PP is follows:

    PQPQPP
    TTFFT
    TFFTT
    FTTFT
    FFTTT

(c)

Expert Solution
Check Mark
To determine

To make: truth table for PQ .

Answer to Problem 3E

    PQPQPQ
    TTFFF
    TFFTT
    FTTFF
    FFTTF

Explanation of Solution

Given information: PQ

Calculation:

The required truth table for PQ is follows:

    PQPQPQ
    TTFFF
    TFFTT
    FTTFF
    FFTTF

(d)

Expert Solution
Check Mark
To determine

To make: truth table for P(Q Q) .

Answer to Problem 3E

    PQPQP(Q Q)
    TTFFT
    TFFTT
    FTTFF
    FFTTF

Explanation of Solution

Given information: P(Q Q)

Calculation:

The required truth table for P(Q Q) is follows:

    PQPQP(Q Q)
    TTFFT
    TFFTT
    FTTFF
    FFTTF

(e)

Expert Solution
Check Mark
To determine

To make: truth table for (PQ)Q .

Answer to Problem 3E

    PQ(PQ)Q
    TTT
    TFT
    FTF
    FFT

Explanation of Solution

Given information: (PQ)Q

Calculation:

The required truth table for (PQ)Q is follows:

    PQP(Q Q)
    TTT
    TFT
    FTF
    FFT

(f)

Expert Solution
Check Mark
To determine

To make: truth table for (PQ) .

Answer to Problem 3E

    PQ(PQ)
    TTF
    TFT
    FTT
    FFT

Explanation of Solution

Given information: (PQ)

Calculation:

The required truth table for PP is follows:

    PQ(PQ)
    TTF
    TFT
    FTT
    FFT

(g)

Expert Solution
Check Mark
To determine

To make: truth table for (PQ) .

Answer to Problem 3E

    PQ(PQ)
    TTF
    TFT
    FTT
    FFT

Explanation of Solution

Given information: (PQ)

Calculation:

The required truth table for (PQ) is follows:

    PQ(PQ)
    TTF
    TFT
    FTT
    FFT

(h)

Expert Solution
Check Mark
To determine

To make: truth table for PQ .

Answer to Problem 3E

    PQPQ
    TTF
    TFF
    FTF
    FFT

Explanation of Solution

Given information: PQ

Calculation:

The required truth table for PQ is follows:

    PQPQ
    TTF
    TFF
    FTF
    FFT

(i)

Expert Solution
Check Mark
To determine

To make: truth table for P(QR) .

Answer to Problem 3E

    PQRP(QR)
    TTTT
    TTFT
    TFTT
    TFFF
    FTTF
    FTFF
    FFTF
    FFFF

Explanation of Solution

Given information: P(QR)

Calculation:

The required truth table for PP is follows:

    PQRP(QR)
    TTTT
    TTFT
    TFTT
    TFFF
    FTTF
    FTFF
    FFTF
    FFFF

(j)

Expert Solution
Check Mark
To determine

To make: truth table for (PQ)(PR) .

Answer to Problem 3E

    PQR(PQ)(PR)
    TTTT
    TTFT
    TFTT
    TFFF
    FTTF
    FTFF
    FFTF
    FFFF

Explanation of Solution

Given information: (PQ)(PR)

Calculation:

The required truth table for (PQ)(PR) is follows:

    PQR(PQ)(PR)
    TTTT
    TTFT
    TFTT
    TFFF
    FTTF
    FTFF
    FFTF
    FFFF

(k)

Expert Solution
Check Mark
To determine

To make: truth table for PP .

Answer to Problem 3E

    PQPP
    TTT
    TFT
    FTF
    FFF

Explanation of Solution

Given information: PP

Calculation:

The required truth table for PP is follows:

    PQPP
    TTT
    TFT
    FTF
    FFF

( l )

Expert Solution
Check Mark
To determine

To make: truth table for (PQ)(RS) .

Answer to Problem 3E

    PQRSPQRS(PQ)(RS)
    TTTTTFT
    TTTFTTT
    TTFTTFT
    TTFFTFT
    TFTTFFF
    TFTFFTT
    TFFTFFF
    TFFFFFF
    FTTTFFF
    FTTFFTT
    FTFTFFF
    FTFFFFF
    FFTTFFF
    FFTFFTT
    FFFTFFF
    FFFFFFF

Explanation of Solution

Given information: Make a truth table for each of the following propositional forms.

Calculation:

The required truth table for (PQ)(RS) is follows:

    PQRSPQRS(PQ)(RS)
    TTTTTFT
    TTTFTTT
    TTFTTFT
    TTFFTFT
    TFTTFFF
    TFTFFTT
    TFFTFFF
    TFFFFFF
    FTTTFFF
    FTTFFTT
    FTFTFFF
    FTFFFFF
    FFTTFFF
    FFTFFTT
    FFFTFFF
    FFFFFFF

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Chapter 1 Solutions

A Transition to Advanced Mathematics

Ch. 1.1 - Give a useful denial of each statement. Assume...Ch. 1.1 - Restore parentheses to these abbreviated...Ch. 1.1 - Other logical connectives between two propositions...Ch. 1.1 - Other logical connectives between two propositions...Ch. 1.2 - Identify the antecedent and the consequent for...Ch. 1.2 - Prob. 2ECh. 1.2 - What can be said about the truth value of Q when...Ch. 1.2 - Identify the antecedent and the consequent for...Ch. 1.2 - Which of the following conditional sentences are...Ch. 1.2 - Which of the following are true? Assume that x and...Ch. 1.2 - Make truth tables for these propositional forms....Ch. 1.2 - Prove Theorem 1.2.2 by constructing truth tables...Ch. 1.2 - Determine whether each statement qualifies as a...Ch. 1.2 - Prob. 10ECh. 1.2 - Dictionaries indicate that the conditional meaning...Ch. 1.2 - Show that the following pairs of statements are...Ch. 1.2 - Prob. 13ECh. 1.2 - Give, if possible, an example of a false...Ch. 1.2 - Give the converse and contrapositive of each...Ch. 1.2 - Prob. 16ECh. 1.2 - The inverse, or opposite, of the conditional...Ch. 1.3 - Translate the following English sentences into...Ch. 1.3 - For each of the propositions in Exercise 1, write...Ch. 1.3 - Translate these definitions from the Appendix into...Ch. 1.3 - Prob. 4ECh. 1.3 - The sentence “People dislike taxes” might be...Ch. 1.3 - Let T={17},U={6},V={24} , and W={2,3,7,26} . In...Ch. 1.3 - (a) Complete the following proof of Theorem...Ch. 1.3 - Which of the following are true? The universe for...Ch. 1.3 - Give an English translation for each. The universe...Ch. 1.3 - Which of the following are true in the universe of...Ch. 1.3 - Let A(x) be an open sentence with variable x. (a)...Ch. 1.3 - Suppose the polynomials anxn+an1xn1+...+a0 and...Ch. 1.3 - Which of the following are denials of (!x)P(x) ?...Ch. 1.3 - Riddle: What is the English translation of the...Ch. 1.4 - Analyze the logical form of each of the following...Ch. 1.4 - A theorem of linear algebra states that if A andB...Ch. 1.4 - Verify that [(BM)L(ML)]B is a tautology. See the...Ch. 1.4 - These facts have been established at a crime...Ch. 1.4 - Prob. 5ECh. 1.4 - Let a and b be real numbers. Prove that (a)...Ch. 1.4 - Suppose a, b, c, and d are integers. Prove that...Ch. 1.4 - Give two proofs that if n is a natural number,...Ch. 1.4 - Let a, b, and c be integers and x, y, and z be...Ch. 1.4 - Recall that except for degenerate cases, the graph...Ch. 1.4 - Exercises throughout the text with this title ask...Ch. 1.5 - Analyze the logical form of each of the following...Ch. 1.5 - A theorem of linear algebra states that if A andB...Ch. 1.5 - Let x, y, and z be integers. Write a proof by...Ch. 1.5 - Write a proof by contraposition to show that for...Ch. 1.5 - A circle has center (2,4) . (a) Prove that (1,5)...Ch. 1.5 - Suppose a and b are positive integers. Write a...Ch. 1.5 - Prob. 7ECh. 1.5 - Prob. 8ECh. 1.5 - Prove by contradiction that if n is a natural...Ch. 1.5 - Prove that 5 is not a rational number.Ch. 1.5 - Three real numbers, x, y, and z, are chosen...Ch. 1.5 - Assign a grade of A (correct), C (partially...Ch. 1.6 - Prove that (a) there exist integers m and n such...Ch. 1.6 - Prove that for all integers a, b, and c, If...Ch. 1.6 - Prove that if every even natural number greater...Ch. 1.6 - Provide either a proof or a counterexample for...Ch. 1.6 - (a) Prove that the natural number x is prime if...Ch. 1.6 - Prove that (a) for every natural number n, 1n1 ....Ch. 1.6 - Starting at 9 a.m. on Monday, a hiker walked at a...Ch. 1.6 - Show by example that each of the following...Ch. 1.6 - Assign a grade of A (correct), C (partially...Ch. 1.7 - (a) Let a be a negative real number. Prove that if...Ch. 1.7 - Prob. 2ECh. 1.7 - Prove that (a) 5n2+3n+4 is even, for all integers...Ch. 1.7 - Prob. 4ECh. 1.7 - Prove that (a) if x + y is irrational, then either...Ch. 1.7 - Prob. 6ECh. 1.7 - Prob. 7ECh. 1.7 - Prob. 8ECh. 1.7 - Prob. 9ECh. 1.7 - Prob. 10ECh. 1.7 - Assign a grade of A (correct), C (partially...Ch. 1.8 - For each given pair a, b of integers, find the...Ch. 1.8 - Prob. 2ECh. 1.8 - Let a and b be integers, a0 , and ab . Prove that...Ch. 1.8 - Prob. 4ECh. 1.8 - Prob. 5ECh. 1.8 - Prob. 6ECh. 1.8 - Prob. 7ECh. 1.8 - Prob. 8ECh. 1.8 - Prove that for every prime p and for all natural...Ch. 1.8 - Let q be a natural number greater than 1 with the...Ch. 1.8 - Prob. 11ECh. 1.8 - Prob. 12ECh. 1.8 - Let a and b be nonzero integers that are...Ch. 1.8 - Let a and b be nonzero integers and d=gcd(a,b) ....Ch. 1.8 - Let a and b be nonzero integers and c be an...Ch. 1.8 - Prob. 16ECh. 1.8 - Prob. 17ECh. 1.8 - Let a and b be integers, and let m=lcm(a,b) . Use...Ch. 1.8 - The greatest common divisor of positive integers a...Ch. 1.8 - Prob. 20ECh. 1.8 - Prob. 21E
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