LMD Async Tutoring

[LMD]

Are you familiar with the distributive property related to addition and multiplication?

Yup, roughly:

4(8) = 8 + 8 + 8 + 8 = (4 x 8)

4(8 + 2) = (8 + 2) + (8 + 2) + (8 + 2) + (8 + 2) = 8 + 8 + 8 + 8 + 2 + 2 + 2 + 2 = (4 x 8) + (4 x 2)

a ∧ (b ∨ c) ≣ (a ∧ b) ∨ (a ∧ c)

Okay, I can see how it could work. I don’t fully understand it though.

The basics of sets are important. That includes terminology like: set, element, subset, union, intersection, empty set, superset. You should know basic operations with sets and be able to answer basic questions like are the elements in a set ordered and can a set contain duplicates?

I’m ~familiar with all those terms except union, intersection, and empty set. My familiarity comes from intuitively picking up the terms from reading philosophy stuff. Like I know that a set is like an abstract grouping, perhaps defined by properties of elements. An element of a set is one of the things that the set contains, that is grouped with other elements which share the properties that define the set. A subset is a set entirely contained within a set. A superset is a set which entirely contains a set.

I think empty set is probably a set with no elements. An intersection maybe a common part of two sets. An intersection… no good guess, the point where two sets meet?

[Elliot]

Look up set basics.

[LMD]

@LMD

An intersection maybe a common part of two sets. An intersection… no good guess, the point where two sets meet?

Oh weird I made a error here with one of these uses of the word intersection. One was meant to be a guess for what a union was and one a guess for what an intersection was. I don’t remember which was which. My thought now is a union is maybe a joining of sets and an intersection is an overlapping of them. I’m about to research this now and wanted to refresh myself on my answers.

[LMD]

Set:

I said:

…a set is like an abstract grouping, perhaps defined by properties of elements.

from SEP:

Sets are well-determined collections that are completely characterized by their elements.

from libretexts:

Intuitively, a set is a collection of objects with certain properties.

from testbook:

A set is a collection of well-defined objects.

Cool yes I think my concept here was right. A collection of objects characterised by the properties of those objects.

---

Element:

I said:

An element of a set is one of the things that the set contains, that is grouped with other elements which share the properties that define the set.

libretexts:

The objects in a set are called the elements or members of the set.

testbook:

The object in a set is called its element or member.

Okay yup. Elements are the things the set contains. The second part of that sentence was kinda redundant.

---

Subset:

I said:

A subset is a set entirely contained within a set.

SEP:

We say that 𝐴 is a subset of 𝐵, written 𝐴⊆𝐵, if every element of 𝐴 is an element of 𝐵.

testbook:

Consider A and B to be two sets. If each element of A is present in set B then A is designated a subset of B

Okay cool. I was right, but I like the description in terms of sharing every element better though.

In the light of that I’d say that a set A is a superset of B if A contains every element in set B.

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Superset:

I said:

A superset is a set which entirely contains a set.

Weird none of those links 3 links above mention ‘superset’ or ‘super set’.

cuemath

Set A is called the superset of another set B if all elements of set B are elements of set A.

Cool okay.

---

Union

SEP:

The set 𝐴 ∪ 𝐵, called the union of 𝐴 and 𝐵, whose elements are the elements of 𝐴 and the elements of 𝐵.

Okay so a union is like the sum of the sets/elements of the sets. It’s all their elements added together. Like set A and set B.

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Intersection

SEP:

The set 𝐴 ∩ 𝐵, called the intersection of 𝐴 and 𝐵, whose elements are the elements common to 𝐴 and 𝐵.

Okay so like overlapping parts of the sets. The common elements between set A and B.

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Empty Set

I said:

a set with no elements

wikipedia:

In mathematics, the empty set is the unique set having no elements; its size or cardinality (count of elements in a set) is zero.[1]

Okay so there is only one empty set. It is not an empty set but the empty set. It’s a unique set. Okay. That kinda makes sense. All sets with no elements are identical to each other.

Edit: forgot some quote attributions

[LMD]

@Elliot

Addition has similarities to which logical operator?

Multiplication has similarities to which logical operator?

Addition is commutative. I.e order doesn’t matter. Both ‘and’ and ‘or’ are like this. Addition feels more like ‘and’ to me. 5 ‘and’ 6 = 11 makes sense.

A and B means both A and B together. Like you’re adding A to B or B to A.

1 add 0 is 1. but with and, 1 and 0 is 0. So it works kind of like multiplication in this respect (1 is the multiplicative identity, and 0 the additive identity)

The additive identity is 0. It can be added to any number without changing it’s identity.

what’s the ‘or’ identity? if you ‘or’ an input with ‘0’, you get the same output as the first input. That’s kind of similar.

Multiplication is commutative. The multiplicative identity is 1. Multiplication scales things. Multiplication can take more than 2 inputs.

What are some logical operators?

and
or
not
xor
equality
implies

Does ‘or’ make sense for either? I don’t think so. I don’t think any of the others do either. not - no, one input. xor… no? equality… hmm no? implies… no, order matters.

I think both addition and multiplication both have similarities to ‘and’. Moreso than others as far as I can see.

[Elliot]

Did you read this post? Eternity Async Tutoring - #371 by Eternity

[LMD]

@Eternity

Additive Identity: A + 0 = A
“Or” Identity: A | 0 = A

Multiplicative Identity: A * 1 = A
“And” Identity: A & 1 = A

Oh okay. I can see this now. I was getting somewhere with the identities. maybe my intuition around adding and 'and’ing meant I didn’t consider that adding could be like 'or’ing.

[Elliot]

look at the distributive property stuff too

[LMD]

@Elliot

Did you read this post? Eternity Async Tutoring - #371 by Eternity

I did, but I kind of ‘lightly’ read it. Hmm. I can see that there are relevant things there I missed. Maybe I was more focused on finding direct prompts from you for me to do as assignments. I don’t know.

@Elliot

look at the distributive property stuff too

Yeah okay, will do.

[LMD]

Okay here is my working on the distributive law. It took a bit of time to wrap my head around. I don’t have something like an intuitive understanding of it, but I can see how it works for now.

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a|b & a|c = 1

This means in english: true if either a or b or both is true and either a or c or both is true.

So if a is true, it’s true. Thats because both ‘or’ scenarios then output true to the ‘and’ so the ‘and’ outputs true.

It’s also true if a and b is true, or a and c is true, but since we know it’s true if just a is true, we can ignore the cases in which a and ‘anything else’ is true.

So if we list the true input scenarios for a|b & a|c separated by ‘or’ we have:

1. a
2. b & c

This means in english: true if a is true, or b and c is true, or both.

which is our a|b&c that a|b & a|c is equivalent to.

a|b & a|c = a|(b&c)

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a&b | a&c = 1

In english this means: true if either a and b, a and c, or both, are true.

This can be looked at as showing us two true input scenarios, either of which will cause a&b | a&c to be true:

1. (a & b)
2. (a & c)

Because a is in both true input scenarios, a has to be true for ‘(a & b) | (a & c)’ to be true. But only either b or c has to be true. In other words: a and (b or c). Or:

a&(b|c)

This is true when a is true while b is true (a & b), or when a is true while c is true (a & c). So we can see that both a&(b|c) and a & b | a & c have the same set of true input cases. This means they are logically equivalent. so:

a&(b|c) = a&b | a&c

[Elliot]

@LMD

Okay here is my working on the distributive law. It took a bit of time to wrap my head around. I don’t have something like an intuitive understanding of it, but I can see how it works for now.

Are you familiar with the distributive law for arithmetic? Compare them.

[LMD]

Yep, roughly.

arithmetic:

a(b + c) = ab + ac

e.g 2(5 + 3) = (5 + 3) + (5 + 3) = 5 + 5 + 3 + 3 = (5 + 5) + 3 + 3 = 2(5) + 2(3)

logic:

a&(b|c) = a|b & a|c
a|(b&c) = a&b | a&c

I can see that these represent similar symbol transformations. But with arithmetic this is intuitively easier for me to understand the equivalence than with logic. I think because with arithmetic I can see clearly how to translate between addition and multiplication. Thinking of the logic ones as being like specifying true inputs scenarios like above has worked, but it’s slow. I probably need more practise with it, or more ways to think about the concept. I can translate ‘or’ to ‘and’ using de morgans laws but the derivations get a little complicated.

[LMD]

@Elliot

Yes. Do you also see the connection with the distributive law?

So A + 0 = A
and A | 0 = A

A * 0 = 0
A & 0 = 0

Do I see the connection with the identity law and the distributive law? Or the connection between the distributive law in arithmetic and logic?

I feel confused. I don’t know if I do or don’t see the connection.

[Elliot]

@LMD

Or the connection between the distributive law in arithmetic and logic?

this one. you said already something about how the symbol transformation was the same. i think that’s good enough for now.

[LMD]

@Elliot

i think that’s good enough for now.

Okay sure.

@Elliot

5x^2 + 3x +11 = 0

We’re not going to solve this, but do you know what the terms on the left are? Can you define a term?

The terms on the left are: 5x^2, 3x, and 11.

Can I define a term? They are the mathematical objects that when grouped together form an expression.

They can be expressed as the product of a coefficient and a variable raised to a power. (Sometimes the coefficient, or variable, or exponent are simplified away like how the term x can be thought of as 1x^1, the term 3x can the be thought of as 3x^1, and the term 11 can be thought of as 11x^0. So each the terms on the left of that equation can be thought of as: 5x^2, 3x^1, 11x^0 respectively.)

[LMD]

@Elliot

What separates terms? What combines things within terms?

Terms are separated by addition(subtraction) and are their parts are combined with multiplication(division).

[LMD]

@Elliot

yes. ok so can you find the terms on the right side of this?

xor(x,y) = (x&~y) | (~x&y)

terms are combined with multiplication, whose analogue in logic is ‘and’, and separated by addition, whose analogue in logic is ‘or’.

So the terms on the right are: (x&!y), (~x&y)

[LMD]

@Elliot

yes. terms will probably come up more later.

You can implement logic operators using arithmetic operators or vice versa. Would you like assignments for that? It’s not necessary but it’s a way to get more practice and it could be interesting or useful.

Okay, no issues. Yes, that sounds good.

@Eternity

I’d like some assignments related to implementing logic operators using arithmetic operators or vice versa.

Cool, I’ll continue with the Eternity topic.

[LMD]

@Elliot

T=1
F=0

and(x,y) = ?
or(x,y) = ?
not(x) = ?

implement with math but no logic operators

hints are available. don’t get stuck for more than around 10min

Ok. I’m not sure what you mean by implement. Maybe write as equivalent arithmetic expressions?:

and(x, y) = x * y

1 * 1 = 1
1 * 0 = 0
0 * 0 = 0
0 * 1 = 0

or(x, y) = x + y

1 + 1 = 1
1 + 0 = 1
0 + 1 = 1
0 + 0 = 0

not(x) = -1?

Hmm… if x is 1 then not(x) is 0, if x is 0 then not(x) is 1…

Stuck on not(x) longer than 10 mins.

[LMD]

@Elliot

First hint:

For each operator, there are essentially 2 different problems with 2 separate solutions. One is for integer arithmetic and one is for real number arithmetic. (In some simpler cases like for “and” there won’t be a difference.)

This reminds me of something I realised earlier. I wonder if this has something to do with it:

@LMD

Interesting so AND and OR do opposite things when the inputs are different. But when the inputs are the same, they both do the same thing, that is the behave like an identity gate. So an AND and OR gate both preserve the inputs when they are equal, but one(AND) outputs false when the inputs are different, and OR outputs true when the inputs are different.

So an AND kind of behaves like an Identity gate and a zero gate. And an OR behaves like an identity gate and a one gate. With equal inputs it functions as an identity gate, and with unequal inputs they function as either a zero(AND) or one(OR) gate.

Hmm idk. you also say that for some simpler cases like for “and” there won’t be a difference.

No, still stuck.