Element-wise Operations

{\displaystyle A + B}

A
# B

∑

a
,
b

(
a
b
)

{\displaystyle A+B=\sum \_{a,b}(a+b)}

a
b

{\displaystyle a+b}

a
# a
a

{\displaystyle a + a = a}

A
⋅
B

{\displaystyle A \cdot B}

A
⋅
# B

∑

a
,
b

a
⋅
b

{\displaystyle A\cdot B=\sum \_{a,b}^{}a\cdot b}

a
⋅
# b
0

{\displaystyle a\cdot b=0}

a
⋅
# a
a

{\displaystyle a\cdot a=a}

A
B

{\displaystyle \frac{A}{B}}

A
B

=

∑

a
′

s

∏

b
′

s

a
b

{\displaystyle {\frac {A}{B}}=\sum \_{a's}\prod \_{b's}{\frac {a}{b}}}

a
b

=
a

{\displaystyle {\frac {a}{b}}=a}

a
a

=
0

{\displaystyle {\frac {a}{a}}=0}

A
Δ
B

{\displaystyle A\Delta B}

a
b

b
a

=

a
b

a
b

{\displaystyle {\frac {a}{b}}+{\frac {b}{a}}={\frac {a+b}{ab}}}

a
b

{\displaystyle a+b}

a
# a
0

{\displaystyle a+a=0}

{\displaystyle x \in A}

x
# A
A

{\displaystyle x+A=A}

x
.
# A
x

{\displaystyle x.A=x}

A
⊆
B

{\displaystyle A \subseteq B}

A
# B
B

{\displaystyle A+B=B}

A
.
# B
A

{\displaystyle A.B=A}

x
∈
A

⇒

x
# A
A

⇒

x
A
# B
A
B

⇒

x
# B
B

⇒

x
∈
B

{\displaystyle x\in A\ \Rightarrow \ x+A=A\ \Rightarrow \ x+A+B=A+B\ \Rightarrow \ x+B=B\ \Rightarrow \ x\in B}

x
∈
A

⇒

x
⋅
# A
x

⇒

x
⋅
A
⋅
# B
x

⇒

x
⋅
# B
x

⇒

x
∈
B

{\displaystyle x\in A\ \Rightarrow \ x\cdot A=x\ \Rightarrow \ x\cdot A\cdot B=x\ \Rightarrow \ x\cdot B=x\ \Rightarrow \ x\in B}

### Three Sets +C

(
A
B
)
C

{\displaystyle (A+B)+C}

A
# B

∑

a
,
b
,
c

(
a
b
c
)

{\displaystyle A+B=\sum \_{a,b,c}(a+b+c)}

(
a
b
)
# c
a
b
c

{\displaystyle (a+b)+c=a+b+c}

A
⋅
B
C

{\displaystyle A\cdot B+C}

∑

a
,
b
,
c

(
a
⋅
b
c
)

{\displaystyle \sum \_{a,b,c}(a\cdot b+c)}

a
⋅
b
# c
(
a
c
)
⋅
(
b
c
)

{\displaystyle a\cdot b+c=(a+c)\cdot (b+c)}

A
B

C

{\displaystyle {\frac {A}{B}}+C}

∑

a
,
c

∏

b

(

a
b

c
)

{\displaystyle \sum \_{a,c}\prod \_{b}({\frac {a}{b}}+c)}

a
b

c

{\displaystyle {\frac {a}{b}}+c}

### Three Sets •C

(
A
B
)
⋅
C

{\displaystyle (A+B)\cdot C}

A
# B

∑

a
,
b
,
c

(
a
b
)
⋅
c

{\displaystyle A+B=\sum \_{a,b,c}(a+b)\cdot c}

(
a
b
)
⋅
# c
a
⋅
c
b
⋅
c

{\displaystyle (a + b) \cdot c = a \cdot c + b \cdot c}

A
⋅
B
⋅
C

{\displaystyle A \cdot B \cdot C}

∑

a
,
b
,
c

a
⋅
b
⋅
c

{\displaystyle \sum \_{a,b,c}a\cdot b\cdot c}

(
a
⋅
b
)
⋅
# c
a
⋅
b
⋅
c

{\displaystyle (a\cdot b)\cdot c=a\cdot b\cdot c}

A
B

⋅
C

{\displaystyle {\frac {A}{B}}\cdot C}

A
⋅
C

B

{\displaystyle {\frac {A\cdot C}{B}}}

a
b

⋅
# c

a
⋅
c

b

{\displaystyle {\frac {a}{b}}\cdot c={\frac {a\cdot c}{b}}}

### Three Sets /C

A
B

C

{\displaystyle {\frac {A+B}{C}}}

A
# B

∑

a
,
b

∏

c

a
b

c

{\displaystyle A+B=\sum \_{a,b}\prod \_{c}{\frac {a+b}{c}}}

a
b

c

=

a
c

b
c

{\displaystyle {\frac {a+b}{c}}={\frac {a}{c}}+{\frac {b}{c}}}

A
⋅
B

C

{\displaystyle {\frac {A\cdot B}{C}}}

A
# B

∑

a
,
b

∏

c

a
⋅
b

c

{\displaystyle A+B=\sum \_{a,b}\prod \_{c}{\frac {a\cdot b}{c}}}

a
⋅
b

c

=

a
c

⋅

b
c

{\displaystyle {\frac {a\cdot b}{c}}={\frac {a}{c}}\cdot {\frac {b}{c}}}

A

/

B

C

{\displaystyle {\frac {A/B}{C}}}

A

B
C

{\displaystyle {\frac {A}{B+C}}}

### Three Sets A/

A

B
C

{\displaystyle {\frac {A}{B+C}}}

∑

a

∏

b
,
c

a

b
c

{\displaystyle \sum \_{a}\prod \_{b,c}{\frac {a}{b+c}}}

a

b
c

=

a
b

⋅

a
c

{\displaystyle {\frac {a}{b+c}}={\frac {a}{b}}\cdot {\frac {a}{c}}}

A

B
⋅
C

{\displaystyle {\frac {A}{B\cdot C}}}

∑

a

∏

b
,
c

a

b
⋅
c

{\displaystyle \sum \_{a}\prod \_{b,c}{\frac {a}{b\cdot c}}}

a

b
⋅
c

=

a
b

a
c

{\displaystyle {\frac {a}{b\cdot c}}={\frac {a}{b}}+{\frac {a}{c}}}

A

B

/

C

{\displaystyle {\frac {A}{B/C}}}

∑

a
,
c

∏

b
′

s

(

a
b

a
⋅
c
)

{\displaystyle \sum \_{a,c}\prod \_{b's}({\frac {a}{b}}+a\cdot c)}

a
b

a
⋅
c

{\displaystyle {\frac {a}{b}}+a\cdot c}

∑

# i
0

n
−
1

i

{\displaystyle \sum\_{i=0}^{n-1} i}