Use rules of inference to show that if ∀x(P(x) ∨Q(x)) and ∀x((P(x) ∧Q(x)) →R(x)) are true, then

Use rules of inference to show that if ∀x(P(x) ∨Q(x)) and...

Key Concepts

Contrapositive Reasoning

Contrapositive reasoning involves proving an implication by showing that the negation of the conclusion implies the negation of the premise. This approach is based on the logical equivalence between an implication and its contrapositive and can simplify proofs where direct reasoning may be more difficult.

Proof by Cases

Proof by cases is a reasoning technique used to handle situations where a statement is known to be true in one of several forms, such as in disjunctive situations. By considering each possible case and showing that the conclusion holds in every instance, one can conclude that the desired result is universally valid.

Universal Quantification

Universal quantification is a fundamental concept in predicate logic that expresses that a given property or relation holds for all elements in a domain. It allows one to take general statements and reason about an arbitrary element, often through techniques like universal instantiation, which is essential for proving statements that themselves are universally quantified.

Rules of Inference

Rules of inference are logical principles that justify the steps taken in a proof. They include methods such as universal instantiation, modus ponens, and others that enable one to derive conclusions from premises in a structured way. These rules provide the rigorous foundation needed for constructing valid logical arguments.

Transcript

00:01 So here we're asked to use rules of inference to show that if two premise that are given are true, then the conclusion is true.

00:10 So let's write our premise and then we'll write our conclusion that are given to us.

00:22 So the premise for all x, p of x or k of x, that's one.

00:38 The next one is for all x, not p of x and q of x, then r of x.

01:05 And the conclusion that's given to us, for every x, not r of x, then p of x.

01:22 All right, so this is going to be a long one, so bear with me.

01:27 So we'll write the step and the reason.

01:31 And we'll go through each step in order to prove that if we assume that these two premise are true, then this conclusion is also true.

01:46 So number one, step one, for every x, p of x, or k of x.

02:02 The reason we use this is because it's a premise.

02:07 Step 2 for every x, not p of x, and q of x, then r of x.

02:30 This is also a premise.

02:36 Now, step three, we'll look at step one, and we assign a c.

02:46 So p of c or q of c, this is universal instantiation using step one, and for step four, we do the same thing but for step 2.

03:03 So we have not p of c and q of c then r of c.

03:23 That's also universal instantiation this time using step 2.

03:32 Now step 5 we look at step 4 and we look at the different logical equivalences that we have and we find one called the conditional disjunction equivalence that's written in this form, if p, then q, we know it's equivalent to not p or q.

03:58 So we could rewrite this as not not p.

04:09 Get rid of that real quick.

04:11 So this is not not p of c and q of c or k or r of c.

04:30 And this is thanks to the conditional disjunction equivalence using step four.

04:45 Now step six, we could write step five, we could rewrite it as not p of c and not q of c or r of c.

05:08 And this is thanks to the morgan's law using step 5.

05:18 So make sure we have it written correctly.

05:20 Yes.

05:27 Quick little error.

05:29 When you use the morgans, the logical connective switches.

05:35 So it goes from a conjunction to a disjunction.

05:41 So step 7, we could see that there is a double negation.