Science:Math Exam Resources/Courses/MATH220/April 2011/Question 04

MATH220 April 2011

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Question 04
Prove that $P\Rightarrow (Q\vee R)\equiv (P\Rightarrow Q)\vee (P\Rightarrow R)$.

Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you?

If you are stuck, check the hints below. Read the first one and consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it! If after a while you are still stuck, go for the next hint.

Hint 1
Use truth tables. Also, remember that $P\Rightarrow Q$ is always true if P is false.

Hint 2
For another solution, do you know or can you prove that $\displaystyle A\Rightarrow B\equiv \lnot A\lor B.$

Checking a solution serves two purposes: helping you if, after having used all the hints, you still are stuck on the problem; or if you have solved the problem and would like to check your work. If you are stuck on a problem: Read the solution slowly and as soon as you feel you could finish the problem on your own, hide it and work on the problem. Come back later to the solution if you are stuck or if you want to check your work. If you want to check your work: Don't only focus on the answer, problems are mostly marked for the work you do, make sure you understand all the steps that were required to complete the problem and see if you made mistakes or forgot some aspects. Your goal is to check that your mental process was correct, not only the result.

Solution 1

Found a typo? Is this solution unclear? Let us know here. Please rate my easiness! It's quick and helps everyone guide their studies. Using truth tables. We write the truth table for each of the two statements that we would like to compare. table for P → (Q ∨ R) P Q R Q ∨ R P → (Q ∨ R) T T T T T T T F T T T F T T T T F F F F F T T T T F T F T T F F T T T F F F F T And table for (P → Q) ∨ (P → R) P Q R P → Q P → R (P → Q) ∨ (P → R) T T T T T T T T F T F T T F T F T T T F F F F F F T T T T T F T F T T T F F T T T T F F F T T T Since the last column is the same for both tables (and since we order the truth of $P$, $Q$ and $R$ in the same order) we can conclude that the two statements are logically equivalent.

Solution 2

Found a typo? Is this solution unclear? Let us know here. Please rate my easiness! It's quick and helps everyone guide their studies. Another solution consists of using an equivalent statement for the implication: $\displaystyle A\Rightarrow B\equiv \lnot A\lor B$ So the statement $\displaystyle P\Rightarrow (Q\lor R)\equiv (P\Rightarrow Q)\lor (P\Rightarrow R)$ becomes $\displaystyle \lnot P\lor (Q\lor R))\equiv (\lnot P\lor Q)\lor (\lnot P\lor R)$ To prove this statement, we'll just show that the left hand side (LHS) is equal to the right hand side (RHS). For the LHS, we get $\displaystyle \lnot P\lor (Q\lor R))\equiv \lnot P\lor Q\lor R$ since disjunction (the logical OR) is associative (that means doesn't care about brackets). For the RHS we get $\displaystyle (\lnot P\lor Q)\lor (\lnot P\lor R)\equiv \lnot P\lor Q\lor R$ since again the disjunction is associative and having twice the ¬P term is useless. We obtained that LHS = RHS which proves the statement.

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