# Can someone break this Propositional logic formulae down for me via truth table

This is my first day learning about logic and logic programming, I have been doing some exercises using a truth table for propositional logic questions

p ∧ (q ∨ r ) ≡ (p ∧ q) ∨ (p ∧ r )


Using a truth table I worked out that this statement is false, but I don't know whether this is correct. If someone could break this down for me or point me to a resource i would really appreciate it.

    p ∧ (q ∨ r ) ≡ (p ∧ q) ∨ (p ∧ r )

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

• You should include the truth table you made for this propositional statement. I suspect that there may be an error in it. – Mike Pierce Jan 18 '16 at 17:46
• @MikePierce I have added the truth table now, although it isn't being pasted correctly – user5647516 Jan 18 '16 at 18:10
• ... really?! Then edit the truth table in your post to make it display correctly. Hint: since there are three symbols in your propositional statement ($p$,$q$,$r$), there should be $2^3=8$ rows in your table. – Mike Pierce Jan 18 '16 at 18:20
• @MikePierce is the above better? Sorry I am just still trying to get my head around this – user5647516 Jan 18 '16 at 18:41
• No, the above is not better. See my answer. – Mike Pierce Jan 18 '16 at 21:21

Here's a truth table for your propositional statement $$p\wedge(q \vee r) \;\;\equiv\;\; (p \wedge q)\vee(p \wedge r)$$ Notice that the last two columns are identical, so the statement is true.

\begin{array}{c|c|c|c|c|c|c|c} p&q&r&(q \vee r)&(p \wedge q)&(p \wedge r)&p\wedge(q \vee r)&(p \wedge q)\vee(p \wedge r)\\\hline T&T&T&T&T&T&T&T\\ T&T&F&T&T&F&T&T\\ T&F&T&T&F&T&T&T\\ T&F&F&F&F&F&F&F\\ F&T&T&T&F&F&F&F\\ F&T&F&T&F&F&F&F\\ F&F&T&T&F&F&F&F\\ F&F&F&F&F&F&F&F\\ \end{array}

• sorry for the late response, thanks it makes sense now @MikePierce – user5647516 Jan 19 '16 at 16:16