Reflexive property - A number is equal to itself.
If x is a real number, then x = x.
Symmetric property - You can reverse two sides of an equation
whenever you want to.
If x = y, then y = x.
Transitive property - You can move from one number through another to
determine a relationship between the numbers.
For equality: If x = y and y = z, then x = z
For order (inequality): If x < y and y < z, then x < z
If x > y and y > z, then x > z
Trichotomy - If x and y are real numbers, then exactly one of the
following must be true:
x<y
x>y
x=y
Addition Property of Equality
Prove that if x = y, then x + z = y + z.
Statements
Reasons
1. x = y
1. Hypothesis (Given)
2. x + z = x + z
3. x + z = y + z
2. Reflexive Property
3. Substitution
Converse of the Addition Property of Equality
Prove that if x + z = y + z, then x = y.
Statements
Reasons
1. x + z = y + z
1. Hypothesis (Given)
2. (x + z) + (-z) = (y + z) + (-z)
3. x + (z + (-z)) = y + (z + (-z))
2. Addition property of equality
3. Associativity for addition
4. x + 0 = y + 0
4. Additive Inverse
5. x = y
5. Additive identity
Cancellation Property of Equality for Addition
Since the addition property of equality is used to prove its converse, the
addition property of equality is a lemma.
A lemma is a theorem that is used to prove a subsequent theorem.
Converse of the Addition Property of Equality
Prove that if x + z = y + z, then x = y.
Statements
Reasons
1. x + z = y + z
1. Hypothesis (Given)
2. (x + z) + (-z) = (y + z) + (-z)
3. x + (z + (-z)) = y + (z + (-z))
2. Addition property of equality
3. Associativity for addition
4. x + 0 = y + 0
4. Additive Inverse
5. x = y
5. Additive identity
Cancellation Property of Equality for Addition
Since the converse of the addition property of equality is proved from
the the addition property of equality, the converse is a corollary.
A corollary is a theorem that is proved by a previous theorem.
If both the theorem and its converse are true, both can be written as a
single statement. x + z = y + z if and only if x = y
Multiplication Property of Equality
Prove that if x = y, then xz = yz.
Statements
Reasons
1. x = y
1. Hypothesis (Given)
2. xz = xz
3. xz = yz
2. Reflexive Property
3. Substitution
Converse of the Multiplication Property of Equality
Prove that if xz = yz, then x = y.
This is false.
If z = 0, then
3•0=5•0
But 3 ≠ 5
Multiplication Property of Negative One
Prove that -1 • x = -x.
Statements
Reasons
-1 • x + x
1. = -1 • x + 1 • x
2. = (-1 + 1)x
1. Multiplicative identity
2. Reverse Distributivity
3. = 0 • x
3. Additive inverse
4. = 0
4. Multiplication Property of zero
5. = -x + x
5. Additive inverse
6. ∴ -1 • x + x = -x + x
6. Transitivity
7. ∴ -1 • x = -x
7. Converse of the addition
property of equality