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A few really really interesting and fun Predicate Problems!

  • 09-04-2012 1:15am
    #1
    mr_november
    Registered Users, Registered Users 2 Posts: 86 ✭✭


    I don't suppose anyone could give me a bit of help with some difficulties I have with a few Predicate problems...well if you can help and wouldn't mind, I'd be very very greatful! :o

    Q1. If the universe of discourse consists o the integers, what are the truth values of the following propositions:

    (a): ∃!x P(x) → ∃xP(x)

    (b): ∃!x ¬P(x) → ¬∀xP(x)

    Attempt: I'm fairly certain both of these are true but that's just because they make sense, I've no way of showing how I came to these answers so maybe they're wrong...:(

    Q2. In the following prroblems, assume the universe of discourse is the set of natural numbers. Using only the standars arithmetic operators and typically mthematical predicaes (=, <, ≤, ≥, >, etc.), express the following athematical concepts using the notation of predicate calculus:

    (a): P(n): n is a prime number.

    (b): There is no largest prime number.

    Attempt:
    (a) Let M(a,b): (a×b=n) and T(a): a>1

    P(n): ¬∃a [T(a)∧T(b)∧M(a,b)]


    (b): P(n): n is a prime (as above) and G(a,b): a< b

    ∀n(P(n) → ∃m [P(m)∧G(n,m)]

    Really not certain about the use of '→' in (b) but can't think of anything else...

    I would be forever greatful if someone could point me in the right direction with these. Thanks if you do! :)
    Tagged:


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