Mathematical Language

The compound statement "$A$ and $B$ are true" is equivalent to "both of $A$ and $B$ are true".

If $A$ or if $B$ or if both are false, then the statement "$A$ and $B$ are true" is false. The statement "$x2 = 4$ and $x > 0$" is true when $x = 2$ and is false for every other value of $x$.

The compound statement "$A$ or $B$ is true" is equivalent to "at least one of $A$ or $B$ is true".

If both $A$ and $B$ are false, then the statement "$A$ or $B$ is true" is false. The statement "$x^2 = 4$ or $x > 0$" is true if $x = –2$ or $x$ is any positive number. The statement is false when $x = –3$ and for lots of other values of $x$.

Practice 1: Which values of $x$ make each statement true?
(a) "$x < 5$"

(b) "$x + 2 = 6$"

(c) "$x^2 – 10x + 24 = 0$"

(d) "$(a)$ and $(b)$"

(e) "$(a)$ or $(c)$"