hello please help me with my homework. I have not received any help

Question

Construct a truth table for the given statement. Please put T for True and F for False. Also, there are many columns, so please do not merge them, but separate them because it can get really confusing.

[(p → q) ∨ (r ∧ ~p)] → (r ∨ ~q)

Answer 1

Looking at our truth table here, let us first identify the meanings of the symbols:

∨ - "OR" So, if either P OR Q is True, we know that the Truth Table for that entry will be T

∧ - "AND" So, here BOTH P AND Q must be true to put a T in the Truth Table entry so even if just 1 of the values are False, then we will have False here.

~ - "NOT" This symbol means Not therefore when it is used, we will have the OPPOSITE OF THE given value so in the case that P is True then ~P will be logically saying Not True, therefore FALSE; similarly, if P was False, then Not False would be True.

→ - The arrow indicates an IF/THEN statement; here if we have a situation where the IF statement is FALSE THEN No matter whether the THEN statement is True or False, we will have a TRUE Statement, I know this seems odd, but if we are starting with something False, then no matter the outcome, we will have a True result. I will break down the 4 possible situations below using the IF/THEN:

True → True: Since we are starting with a True Statement and ending with a True statement, this will be True.

True → False: Since we are starting with a True Statement, however, we have changed the outcome to a False result, this statement will be False.

False → True: Like I mentioned above, if we start with a False statement, our IF/THEN Result will be True.

False → False: Like I mentioned above, if we start with a False statement, our IF/THEN Result will be True. We can also make sense of this because if we are starting with something False it would make sense that we would output something False.

Symbolically,

If P THEN Q

P | Q | P→Q

T| T| T

T|F|F

F|T|T

F|F|T

Now, going through your Truth Table Noting the above (although it is a little confusing that there is ? symbols under the → symbols)

But, given [(p → q) ∨ (r ∧ ~p)] → (r ∨ ~q) and the Following Table:

Please note I am running the above truth table done in steps based on where the parentheses are so for clarity:

P | Q | R | (p → q) | (r ∧ ~p) | [(p → q) ∨ (r ∧ ~p)] | (r ∨ ~q) | [(p → q) ∨ (r ∧ ~p)] → (r ∨ ~q)

T | T | T | (T → T) | (T ∧ ~ T)

(so T) | (so F) | (T ∨ F) |(T ∨ ~ T)

| (so T) | (so T) | (T → T)

(so T)

T | T | F | (T → T) | (F ∧ ~T)

(so T) | (so F) | (T ∨ F) |(F ∨ ~ T)

(so T) | (so F) | (T → F)

|(so F)

T | F | T | (T → F) | (T ∧ ~ T)

(so F) | (so F) | (F ∨ F) |(T ∨ ~ T)

(so F) | (so T) | (F → T)

|(so T)

T | F | F | (T → F) | (F ∧ ~ T)

(so F) | (so F) | (F ∨ F) |(F ∨ ~ T)

(so F) | (so F) | (F→ F)

| (so T)

F | T | T | (F → T) | (T ∧ ~ F)

(so T) | (so T) | (T ∨ T) |(T ∨ ~ F)

(so T) | (so T) | (T → T)

| (so T)

F | T | F | (F → T) | (F ∧ ~ F)

(so T) | (so F) | (T ∨ F) |(F ∨ ~ F)

(so T) (so T) | (T → T)

| (so T)

F | F | T | (F → F) | (T ∧ ~ F)

(so T) | (so T) | (T ∨ T) |((T ∨ ~ F)

(so T) |(so T) | (T → T)

| (so T)

F | F | F | (F → F) | (F ∧ ~ F)

(so T) | (so F) | (T ∨ F) |(F ∨ ~ F)

(so T) |(so T) | (T → T)

| (so T)

I hope this helps! Thanks for reaching out!