Given p is true, q is false, and r is trus, find the truth value of the statement (~p^q)<->~r. Show step by step work.
To find the truth value of the statement (~p^q)<->~r, we need to evaluate the truth values of each component and follow the rules of logical operations. Here are the step-by-step work:
Step 1: Evaluate ~p which means negating the truth value of p.
Since p is given as true, ~p will be false.
Step 2: Evaluate ~r which means negating the truth value of r.
Since r is given as true, ~r will be false.
Step 3: Evaluate (~p^q).
Using the truth values from step 1 and the given truth value of q as false, we can use the logical operation AND (^) to combine the truth values.
~p = false and q = false, so (~p^q) will be false.
Step 4: Evaluate the entire statement (~p^q)<->~r.
Using the truth value from step 2, we can now apply the logical operation BICONDITIONAL (<->) between (~p^q) and ~r.
(~p^q) = false and ~r = false, so (~p^q)<->~r will be true.
Therefore, the truth value of the statement (~p^q)<->~r is true.