My homework is due and i need help with it please!

Question

1.) Determine whether the sentence is a statement.

x = x + 5

A.) The sentence is a statement.

B.) The sentence is not a statement.    

2.) Determine the truth value of the compound statement given that p is a false statement, q is a false statement, and r is a true statement.

[~(p ∧ ~q) ∨ r] ∧ (p ∨ ~r)

A.) True

B.) False    

Please help

Answer 1

A mathematical statement is something that has a clear definition that cannot be interpreted any other way and is ether always true or always false, no matter what values are used and what situation is occurring.

1. In our case, x = x + 5 is clear and straightforward equation. Just given the formula, you know that you have to add 5 without needing to question if it means something else.

The second criteria is it must be unconditionally true or unconditionally false. We can see that it is false because no number on the real number line can make this equation true.

With both criteria met, we can safely say that this equation is a statement.

2. We are given the statement [~(p ∧ ~q) ∨ r] ∧ (p ∨ ~r) and are told that p and q are false while r is true. As for the symbols

~ means 'not', which changes if the variable is true or false

∧ means 'and'. When between two variables, even if just one variable is false, the whole expression will be false

v means 'or'. When between two variables, only one needs to be true for the expression to be true.

Follow the parentheses to know what variables to work with first.

[~(p ∧ ~q) ∨ r] ∧ (p ∨ ~r)

p ∧ ~q: is p and not q. p is false, and the ~ symbol changes q true. We are left with a false and a true between an and operator. That means this part will be overall false.

[~(p ∧ ~q) ∨ r] : the ~ symbol in front of (p ∧ ~q) changes the false discussed earlier into a true, leaving us with a true on both sides of the or symbol, making the over all expression true.

(p ∨ ~r) : p is false and r is true. We have ~r or not r, changing it to false. we are left with false on both sides of the or symbol, making it false altogether.

[~(p ∧ ~q) ∨ r] ∧ (p ∨ ~r). When we put it all together, we are left with a false and true between an and operator. This makes the over all expression false.

Hope that helps.