A Transition To Mathematics With Proofs Cullinane Download Torrent

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  • Step 1 of 20

    The objective is to determine whether the given statements arepropositions and to provide the truth value of eachproposition.

    A sentence that has exactly one truth value is called asProposition. The truth value is either true, denoted by T,or false denoted by F.

  • Step 2 of 20

    a)

    Consider the sentence “What time is dinner?”

    This is not a proposition because it is a question.

  • Step 3 of 20

    b)

    Consider the sentence “It is not the case that is not a rational number.”

    As the given sentence has exactly one truth value, so thesentence is a proposition.

    The sentence is the negation of is not a rational number, thus the sentence is is a rational number.

    But observe that, is an irrational number.

    Thus, the truth value of the sentence is False.

  • Step 4 of 20

    c)

    Consider the sentence “isa rational number.”

    Observe that the truth value of the sentence varies with thevalues of . This is not a proposition.

    The truth value of the sentence can be determined asfollows:

    Suppose

    Then the value of is,

    Thus, and 1 is a rational number.

    Therefore, the truth value is true.

    • is not a proposition. However, it doesn't have a truth value. See the definition of proposition.
    • example, let ×=pi then it will be irrational.
  • Step 5 of 20

    Suppose

    Then the value of is,

    Thus, and is an irrational number.

    Therefore, the truth value is false.

    Thus, the sentence has more than one truth value.

    Hence, the sentence is not a proposition.

  • Step 6 of 20

    d)

    Consider the sentence “isa real number.”

    Observe that the truth value of the sentence varies with thevalues of and . This is not a proposition.

    The truth value of the sentence can be determined asfollows:

    Suppose

    Then the value of is,

    Thus, and 13 is a real number.

    Therefore, the truth value is true.

  • Step 7 of 20

    Suppose

    Then the value of is,

    Thus, and is a complex number.

    Therefore, the truth value is false.

    Thus, the sentence has more than one truth value.

  • Step 8 of 20
  • Step 9 of 20

    e)

    Consider the sentence “Either is rational and 17 is a prime, or and 81 is a perfect square.

    This is a proportion.

    The truth value of the sentence can be determined asfollows:

    Let is rational.

    Then the proposition is equal to

  • Step 10 of 20

    As is rational, so P is false.

    Therefore, is false.

    As , so R and S are true.

    Therefore, is true.

    Thus, the truth value of is true.

    As the sentence has exactly one truth value, so the sentence isa proposition and its truth value is True.

  • Step 11 of 20

    f)

    Consider the sentence “Either 2 is rational and is irrational, or is rational.

    This is a proportion.

    The truth value of the sentence can be determined asfollows:

    Let is rational.

    Then the proposition is equal to

  • Step 12 of 20

    As is rational, so P is true.

    As is irrational, so Q is true.

    Therefore, is true.

    As , so R is false.

    Thus, the value of is true.

    Therefore, the truth value of the proposition istrue.

    As the sentence has exactly one truth value, so the sentence isa proposition and its truth value is True.

  • Step 13 of 20

    g)

    Consider the sentence “Either is rational and is rational, or there are exactly four primes less than 10.

    This is a proportion.

    The truth value of the sentence can be determined asfollows:

    Let is rational.

    Then the proposition is equal to

  • Step 14 of 20

    As is rational, so P is false.

    As is rational, so Q is true.

    Therefore, is false.

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    As , so R is true.

    Thus, the value of is true.

  • Step 15 of 20
    Therefore, the truth value of the propositionis true.

    As the sentence has exactly one truth value, so the sentence isa proposition and its truth value is True.

  • Step 16 of 20

    h)

    Consider the sentence “is rational, and either ”

    This is a proportion.

    The truth value of the sentence can be determined asfollows:

    Let is rational.

    Then the proposition is equal to

  • Step 17 of 20

    As is rational, so P is true.

    As , so Q is true.

    As , so R is false.

    Thus, the value of is true.

    Therefore, the truth value of the proposition istrue.

    As the sentence has exactly one truth value, so the sentence isa proposition and its truth value is True.

  • Step 18 of 20

    i)

    Consider the sentence “It is not the case that 39 is prime, orthat 64 is a power of 2.”

    This is not a proportion.

    The truth value of the sentence can be determined asfollows:

    Let is prime.

    is a power of 2.

    Then the proposition is equal to

  • Step 19 of 20

    As is prime, so P is false.

    Asis a power of 2, so Q is true.

    Therefore, is true.

    Thus, the value of is false.

    Therefore, the truth value of the proposition isfalse.

    As the sentence has exactly one truth value, so the sentence isa proposition and its truth value is False.

  • Step 20 of 20

    j)

    Consider the sentence “There are more than three falsestatements in this book, and this statement is one of them.”

    This is not a proportion.

    The sentence is a paradox.

    Hence, the sentence is not a proportion.

    • proposition is spelt wrong

Feb 07, 2017  Mathematical Proofs A Transition to Advanced Mathematics 3rd Edition Featured Titles for Transition. A transition to mathematics with proofs by Michael J. Cullinane Published 2013 by Jones & Bartlett Learning in Burlington, Mass.

Chapter 1

Logic And Proofs

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1-1Prepositions and ConnectivesExercisesp.7
1-2Conditionals and BiconditionalsExercisesp.15
1-3Quantified StatementsExercisesp.26
1-4Basic Proofs Method IExercisesp.38
1-5Basic Proof Methods IIExercisesp.48
1-6Proofs Involving QuantifiersExercisesp.60
1-7Strategies for Constructing ProofsExercisesp.72
1-8Proofs from number TheoryExercisesp.81

Chapter 2

Sets And Induction

2-1Basic Concepts of Set TheoryExercisesp.91
2-2Set OperationsExercisesp.101
2-3Indexed Families of SetsExercisesp.110
2-4Mathematical InductionExercisesp.123
2-5Equivalent Forms of InductionExercisesp.133
2-6Principles of CountingExercisesp.147

Chapter 3

Relations And Partitions

3-1RelationsExercisesp.160
3-2Equivalence RelationsExercisesp.171
3-3PartitionsExercisesp.179
3-4Modular ArithmeticExercisesp.186
3-5Ordering RelationsExercisesp.196

Chapter 4

A Transition To Mathematics With Proofs Cullinane Download Torrent Pdf

Functions

4-1Functions as RelationsExercisesp.209
4-2Construction of FunctionsExercisesp.218
4-3Functions That are Onto; One-to-One FunctionsExercisesp.227
4-4Inverse FunctionsExercisesp.236
4-5Set ImagesExercisesp.241
4-6SequencesExercisesp.248
4-7Limits and Continuity of Real FunctionsExercisesp.257

A Transition To Mathematics With Proofs Cullinane Download Torrent Free

Transition

Chapter 5

Cardinality

5-1Equivalent Sets; Finite SetsExercisesp.268
5-2Infinite SetsExercisesp.277
5-3Countable SetsExercisesp.284
5-4The Ordering of Cardinal NumbersExercisesp.293
5-5Comparability and the Axiom of ChoiceExercisesp.300

Chapter 6

Concepts Of Algebra

6-1Algebraic StructuresExercisesp.309
6-2GroupsExercisesp.317
6-3SubgroupsExercisesp.324
6-4Operation Preserving MapsExercisesp.331
6-5Rings and FieldsExercisesp.339

Chapter 7

Concepts Of Analysis

7-1The Completeness PropertyExercisesp.349
7-2The Heine-Borel TheoremExercisesp.360
7-3The Bolzano-Weierstrass TheoremExercisesp.366
7-4The Bounded Monotone Sequence TheoremExercisesp.372
7-5Equivalents of CompletenessExercisesp.377