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

1. ## 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

2. ## Re: How would I prove the following are logically equivalent?

Originally Posted by bee77
(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.

3. ## 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

4. ## 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

5. ## Re: How would I prove the following are logically equivalent?

Originally Posted by bee77
(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.

6. ## 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 ....