7 consider the following proposition for each integer a a 2...

Consider the following proposition: For each integer a, a ≡ 2 (mod 8) if and only if (a^2 + 4a) ≡ 4 (mod 8). a) Write the proposition as the conjunction of two conditional statements. b) Determine if the two conditional statements in Part (a) are true or false. If a conditional statement is true, write a proof, and if it is false, provide a counterexample. Is the given proposition true or false? Explain.

Transcript

00:01 So in this question we're asked to explore a bit what it means for equalities modulo 8.

00:10 But first it's just a logical question.

00:12 So essentially we're going to investigate the following statement in the equation that says that for each integer a, a number is equal to 2 mod 8.

00:25 If and only if, if it's square plus 4 times itself, is equal.

00:34 To 4 mod 8.

00:38 And the first part of the question is simply, okay, this is an if and only if, just asks us to explore what if and only if means.

00:45 And it says, write the above as a conjunction of two conditionals.

01:00 All right, so let me just quickly recap what the language means.

01:04 So conjunction is a statement that connect, two statements connect with an and, so something and something.

01:10 And a conditional is a statement of the form.

01:12 If something then something else right so two statements connected by the if and then and then what the other for the end right and we just have to recall that if and only if is an equivalence and then equivalence is an implies and is implied by right so the statement that we want to say is equivalent so the above is equivalent to the statement so a equals to mod 8 implies a squared plus 4a is equal to 4 mod 8.

01:57 And so this is the first conditional and then we do a conjunction and the opposite is also true.

02:04 A squared plus 4a is equal to 4 mod 8 implies a equals to mod 8.

02:14 All right so this is how we write an equivalence as a conjunction of two conditionals right and now part b just says check if each if these are true are true and in this in this case there is going to be one that's easier to prove the other less easy so we have to think and we have to think a bit before doing so let's start with the first one maybe first one is the easy one so we assume that a is equal to two mod eight and then do the calculation to check so suppose that a is equal to 2, mod 8.

02:54 Then let's just make the calculation.

02:56 So a squared plus 4a.

02:58 So a is 2, a squared is 4.

03:01 A is 2.

03:03 So 4 times a is 8, but 8 is equal to 0 if we're working mod 8.

03:08 So this is 4, mod 8, as we wanted to check.

03:12 So the first conditional is true.

03:23 Now for the second conditional, i would suggest the following.

03:28 It's a bit more difficult to choose, right? so now we're going to assume that a squared plus 4a is equal to 4, and then we need to get a conclusion for a...

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