Proofs
A proof is a series of claims that lead to a conclusion. Some proofs are conditional, which means that the claims can only be made under certain conditions. Click on a statement to see the proof
if the following are true:
- a
- b = a
then b
if the following are true:
- a = c
- b = c
then a = b
if the following are true:
- a = b
- a = c
then b = c
if the following are true:
- a = b
- b = c
then c = a
if the following are true:
- a = b
- c = a
then b = c
if the following are true:
- a = b
- b = c
- c = d
then a = d
if the following are true:
- a = b
- a = c
- b = d
then c = d
if the following are true:
- a = x
- b = y
- x = y
then a = b
0 ⋅ a = 0
(1 / a) ⋅ a = 1
(a ⋅ b) ⋅ c = (c ⋅ a) ⋅ b
(b / c) ⋅ c = b
if a ⋅ 1 = b, then a = b
if a = b / c, then c ⋅ a = b
if a + a = b, then a ⋅ 2 = b
if the following are true:
- a = b
- c = d
then a - c = b - d
if a ⋅ 2 = b, then a = b ⋅ (1 / 2)
if a + a = b, then a = b ⋅ (1 / 2)
if (a + b) + c = d, then (a + c) + b = d
if (a + b) + c = d, then (b + a) + c = d
if the following are true:
- x = c + d
- y = e + f
then ((a + b) + (c + d)) + (e + f) = ((a + b) + x) + y
if a ⋅ b = c, then a = c / b
if (a + b) + b = 180, then a + (b ⋅ 2) = 180
if x = (0 + (a ⋅ 2)) / 2, then x = a
if x = ((b ⋅ 2) + (a ⋅ 2)) / 2, then x = b + a
if f = a - 0, then f = a
if f = (a + b) - b, then f = a
if the following are true:
- a < b
- c = a
then c < b
if a = b, then x - a = x - b
if the following are true:
- a = b
- f = c - a
then f = c - b
if a = b, then a - c = b - c
if a - b = c, then a = c + b
if a - b = 0, then a = b
if c = a + a, then c = a ⋅ 2
if a / b = c / d, then d ⋅ a = b ⋅ c
((x ⋅ 4) ⋅ 2) - (2 ⋅ 2) = ((x ⋅ 4) ⋅ 2) - 4
((x ⋅ 4) ⋅ 2) - 4 = (x ⋅ 8) - 4
(c ⋅ a) - (c ⋅ b) = c ⋅ (a - b)
(a ⋅ 2) + (a ⋅ (-2)) = 0
((a ⋅ 1) / 2) + ((a ⋅ 1) / 2) = a
(a ⋅ (1 / c)) ⋅ c = a
if the following are true:
- a = b
- b ⋅ c = d
then a ⋅ c = d
if a / c = b / c, then a = b
if a / c = b / c, then b = a
if the following are true:
- a / b = c / d
- a = w
- b = x
- d = z
then w / x = c / z
((a ⋅ b) ⋅ c) ⋅ d = ((b ⋅ d) ⋅ a) ⋅ c
((a / b) ⋅ c) ⋅ b = a ⋅ c
if a / b = c / d, then a ⋅ d = b ⋅ c
if the following are true:
- a ⋅ b = c
- d = b
then a ⋅ d = c
if a = 1, then b / a = b
(a ⋅ 2) / (b ⋅ 2) = a / b
if the following are true:
- a = (b + c) + d
- b = x + y
then a = ((x + y) + c) + d
if the following are true:
- a = ((b + c) + d) + e
- b = x + y
then a = (((x + y) + c) + d) + e
(a ⋅ 2) + (a ⋅ 2) = a ⋅ 4
if 180 = a + 90, then a = 90
if a ⋅ 2 = 180, then a = 90
if a + a = 180, then a = 90
(((-1) ⋅ 3) ⋅ x) + ((3 ⋅ x) + 20) = 20
(((-1) ⋅ 3) ⋅ x) + (4 ⋅ x) = x
if (3 ⋅ x) + 20 = 4 ⋅ x, then 20 = x
if ((a + b) + c) + e = ((a + b) + g) + h, then ((a + b) + c) + e = ((a + g) + b) + h
if (a + b) + c = 180, then a + c = 180 + (b ⋅ (-1))
if the following are true:
- a = b + c
- c = d
then b + d = a
if the following are true:
- a = b
- b = d
- a = c
then d = c
if ((a + b) + a) + b = 360, then a + b = 180
if ((a + a) + a) + a = 360, then a = 90
if a + a = 90, then a = 45
if 60 + (a ⋅ 2) = 180, then a = 60
if the following are true:
- x = 12
- d = (x ⋅ 2) + 1
then d = 25
if the following are true:
- x = 12
- d = x + 13
then d = 25
if the following are true:
- a = 0
- b = e
then a + b = e
if the following are true:
- y = (a + b) / 2
- a = 0
- b = e
then y = e / 2
if x = a - a, then x = 0
if the following are true:
- x = 0 / y
- not (y = 0)
then x = 0
if the following are true:
- b = 0
- a = 0
then b - a = 0
if a + b = 180 - 90, then a + b = 90
if a = 90 - 67, then a = 23
if a = 90 - 23, then a = 67
(a / b) ⋅ d = (d / b) ⋅ a
(a ⋅ x) / x = a
if a / b = c / d, then d / b = c / a
if the following are true:
- a = 1
- a ⋅ a = b
then 1 = b
if a = b, then a ⋅ a = b ⋅ b
if the following are true:
- x = (a ⋅ a) + (b ⋅ b)
- m = a
- n = b
then x = (n ⋅ n) + (m ⋅ m)
if the following are true:
- a ⋅ a = (b ⋅ b) + (c ⋅ c)
- a = x
- b = y
then x ⋅ x = (y ⋅ y) + (c ⋅ c)
((a ⋅ b) ⋅ c) ⋅ d = ((a ⋅ c) ⋅ b) ⋅ d
(s / 2) ⋅ (s / 2) = (s ⋅ s) / 4
if a = b + c, then a - b = c
a + (b ⋅ ((-1) / c)) = a - (b / c)
(s ⋅ s) - ((s ⋅ s) / 4) = (3 / 4) ⋅ (s ⋅ s)
if the following are true:
- a = b / c
- d = b
- c = e
then a = d / e
if the following are true:
- a = b / c
- b = d
- c = e
then a = d / e
if a = b ⋅ c, then a ⋅ a = ((b ⋅ c) ⋅ b) ⋅ c
(s ⋅ s) ⋅ (1 / 4) = (s / 2) ⋅ (s / 2)
square root of ((s ⋅ s) ⋅ (1 / 4)) = s / 2
(3 / 4) ⋅ (s ⋅ s) = 3 ⋅ ((s ⋅ s) ⋅ (1 / 4))
square root of ((3 / 4) ⋅ (s ⋅ s)) = (square root of 3) ⋅ (s / 2)
if a = b ⋅ b, then square root of a = b
s ⋅ (1 / s) = 1
(s / 2) / s = 1 / 2