Basic proofs / axioms

What does the reflexive property state?

A quantity is equal to itself (e.g., $a = a$).

What is the symmetric property of equality?

If a quantity is equal to another, you can swap them (e.g., if $a = b$, then $b = a$).

What does the transitive property signify?

If one quantity equals a second, and the second equals a third, then the first equals the third (e.g., if $a = b$ and $b = c$, then $a = c$).

What is the addition property of equality?

You may add the same number to both sides of an equation and it will remain true (e.g., if $a = b$, then $a + c = b + c$).

What does the subtraction property of equality state?

You may subtract the same number from both sides of an equation and it will remain true (e.g., if $a = b$, then $a - c = b - c$).

What is the multiplication property of equality?

You may multiply the same number on both sides of an equation and it will remain true (e.g., if $a = b$, then $a \times c = b \times c$).

What does the division property of equality state?

You may divide the same number on both sides of an equation and it will remain true (e.g., if $a = b$ and $c \neq 0$, then $\frac{a}{c} = \frac{b}{c}$).

What is the distributive property?

Multiplication can be distributed across addition or subtraction (e.g., $a(b + c) = ab + ac$).

What does the commutative property allow you to do?

You may change the order of numbers without changing the result (e.g., $a + b = b + a$ or $a \times b = b \times a$).

What is the associative property?

You may change the grouping of numbers when adding or multiplying without changing the result (e.g., $(a + b) + c = a + (b + c)$).

What does the identity property refer to?

Adding zero or multiplying by one leaves the number unchanged (e.g., $a + 0 = a$ or $a \times 1 = a$).

What is the inverse property?

A number combined with its opposite is $0$ for addition; for multiplication, it is the number that multiplies it to one (e.g., $a + (-a) = 0$ or $x \times \frac{1}{x} = 1$).