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Thread: How would I prove the following are logically equivalent?

  1. #1
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    How would I prove the following are logically equivalent?

    (a) (p ∨ q ) and p ∧ q ?
    Which I believe read "not" (p inclusive or q) and not p and q ?
    Would I do a truth table to try and prove it ?

    Also

    (b) (p → q ) and p ∧ q ?

    Which reads "not"(p if q) and p and q ?
    Again would a truth table be used ?
    any help would be appreciated .
    Thanks
    Last edited by bee77; Aug 8th 2017 at 03:45 AM.
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  2. #2
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    Re: How would I prove the following are logically equivalent?

    Quote Originally Posted by bee77 View Post
    (a) (p ∨ q ) and p ∧ q ?
    Which I believe read "not" (p inclusive or q) and not p and q ?
    Would I do a truth table to try and prove it ?

    Also

    (b) (p → q ) and p ∧ q ?

    Which reads "not"(p if q) and p and q ?
    Again would a truth table be used ?
    any help would be appreciated .
    Yes. In both cases use a truth table.
    Thanks from bee77
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  3. #3
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    Re: How would I prove the following are logically equivalent?

    Ok thanks Plato
    I'll have a go at it tomorrow and post my response as I'm off to bed thanks
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  4. #4
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    Re: How would I prove the following are logically equivalent?

    (a) (p ∨ q ) and p ∧ q

    for a the truth table

    | p q | (p ∨ q ) | p ∧ q |
    | T F | F | F |
    | T F | F | F |
    | F T | F | F |
    | F F | T | T |


    We find that using a truth table for a) that (p ∨ q ) and p ∧ q are logically equivalent using the definition of Conjunction ,Disjunction and Negation with Binary Operations on propositions .
    de Morgan’s Law:(p ∨ q ) ≡ p ∧ q also holds true .

    b) (p → q ) and p ∧ q

    for a the truth table

    | p q | (p → q) | p ∧ q |
    | T F | F | F |
    | T F | T | T |
    | F T | F | F |
    | F F | F | F |

    We find that using a truth table for b) that (p → q ) and p ∧ q are logically equivalent using the definition of Conjunction ,Disjunction and Negation with Binary Operations on propositions .
    de Morgan’s Law: (p → q )≡ p ∧ q also holds true .
    Is that correct ?
    Also is my statement ok ?
    Thanks
    Last edited by bee77; Aug 8th 2017 at 04:51 PM.
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  5. #5
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    Re: How would I prove the following are logically equivalent?

    Quote Originally Posted by bee77 View Post
    (a) (p ∨ q ) and p ∧ q

    for a the truth table

    | p q | (p ∨ q ) | p ∧ q |
    | T F | F | F |
    | T F | F | F |
    | F T | F | F |
    | F F | T | T |


    We find that using a truth table for a) that (p ∨ q ) and p ∧ q are logically equivalent using the definition of Conjunction ,Disjunction and Negation with Binary Operations on propositions .
    de Morgan’s Law:(p ∨ q ) ≡ p ∧ q also holds true .

    b) (p → q ) and p ∧ q

    for a the truth table

    | p q | (p → q) | p ∧ q |
    | T F | F | F |
    | T F | T | T |
    | F T | F | F |
    | F F | F | F |

    We find that using a truth table for b) that (p → q ) and p ∧ q are logically equivalent using the definition of Conjunction ,Disjunction and Negation with Binary Operations on propositions .
    de Morgan’s Law: (p → q )≡ p ∧ q also holds true .
    Is that correct ?
    Well done!
    BUT you must simply learn these so that using them in other proofs is second nature to you.
    Thanks from bee77
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  6. #6
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    Re: How would I prove the following are logically equivalent?

    thanks , Plato I've been learning them as I sit here ..the compund ones I'm up to now are tricky ....
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