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Predicate Logic and Relations on sets

  1. Mar 14, 2009 #1
    1. The problem statement, all variables and given/known data

    Here are a few questions from an exercise sheet that I need help on. I really don't have a clue on how to start them. Could anyone help me attempt at each a) for each question?

    1. Use (nested) quantifiers (∀ and ∃) (and propositional junctors) and only equality ``='', conventional ordering ``≤'', ``<'', etc., and divisibility ``|'' as predicates, and arithmetic operators as functions to express the following statements as predicate logic formulae:
    a)**“Exponentiation on reals distributes over multiplication to the left.”
    b)**“Each positive real number has a logarithm.”
    c)**“Exponentiation on reals has no left identity.”
    d)**“Any two natural numbers have a least common multiple.”
    e)**“There are infinitely many primes. (I.e., for each natural number, there is a prime number above it.)”
    f)**“Each Pythagorean triple involves at least one even number.”

    2. Assume that Q and R are relations on a set A. Prove (using relation-algebraic calculations) or disprove (by providing counterexamples) each of the following statements.
    a)**If Q and R are both reflexive, then Q ∩ R is reflexive, too.
    b)**If Q and R are both reflexive, then Q ∪ R is reflexive, too.
    c)**If Q and R are both transitive, then Q ; R is transitive, too.
    d)**If Q and R are both symmetric, then Q ∩ R is symmetric, too.
    e)**If Q and R are both transitive, then Q ∪ R is transitive, too.
    f)**If Q is symmetric, then Q ; Q is symmetric, too.

    3. The attempt at a solution
    1a) ∀x∀y∀z((xy)z = xzyz | x,y,z ∈ ℝ)
    2a) not sure how to start...

    Many thanks.
  2. jcsd
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