A prove that p mid q leftrightarrow negp wedge q b prove...

Name: a prove that p mid q leftrightarrow negp wedge q b prove that neg p leftrightarrow p mid p c build wedge using only the sheffer stroke d build vee using only the sheffer stroke 01974
Uploaded: 2023-06-22T15:10:45-08:00
Duration: 3 min 5 s
Channel: N D
Description: a prove that p mid q leftrightarrow negp wedge q b prove that neg p leftrightarrow p mid p c build wedge using only the sheffer stroke d build vee using only the sheffer stroke 01974

(a) Prove that $p \mid q \Leftrightarrow \neg(p \wedge q)$. (b) Prove that $\neg p \Leftrightarrow p \mid p$. (c) Build $\wedge$ using only the Sheffer Stroke. (d) Build $\vee$ using only the Sheffer Stroke.

Transcript

00:01 In this problem, we're looking at a new type of logical operator called n and.

00:06 So this right here is using this new operator, which we called n and.

00:13 So we would say p, n, and q there.

00:15 And we want to show that this is logically equivalent to not p and q.

00:21 So we can do this using a truth table.

00:23 We want to show that they're going to have the same truth values.

00:27 And what we can do is first fill in the column for p and q, making sure we have every single combination of true and false.

00:35 So i'm going to do for p, true, true, false, false.

00:40 And then for q, i'm going to do true, false, true, false.

00:45 So then we can determine this column of p, n, and q.

00:49 And then we can determine not p and q by first determining what p and q is, and then negating those truth values.

00:56 And then we just want to compare those two columns to make sure that they have the same trues and falses in the same spots.

01:02 So actually, this p, n, and q, finding the truth values for this came from a previous problem, but let's look at it again.

01:13 So the definition of p, n, and q is that it's true when either p or q or both are false.

01:22 And it's false when both p and q are true.

01:26 So if both p and q are true, like it is in this first column, then our p, n, and q is false.

01:34 And it's going to be true when there's at least one false.

01:37 So for the rest of these, it's going to be true...

Question

(a) Prove that $p \mid q \Leftrightarrow \neg(p \wedge q)$. (b) Prove that $\neg p \Leftrightarrow p \mid p$. (c) Build $\wedge$ using only the Sheffer Stroke. (d) Build $\vee$ using only the Sheffer Stroke.

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