Logic: If–Then, AND, OR, NOT

Logic decides whether statements are True or False. AND, OR and NOT combine them, and Venn diagrams count groups that overlap.

Key ideas

A statement is either True or False; AND, OR and NOT build bigger statements from smaller ones.
P AND Q is True only when both are True; P OR Q is True when at least one is True.
NOT turns True into False and False into True; work out brackets first.
An if–then rule is broken only by a case where the “if” part holds but the “then” part does not.
In a Venn diagram, people in both groups sit in the overlap and must be counted once.
A rule learned from labelled examples must fit all of them.

Remember

AND	True only if both are True
OR	False only if both are False
NOT	NOT True = False; NOT False = True
At least one group	A + B − both
Neither	total − (A + B − both)
Example	30 children, 18 like tea, 12 like milk, 5 both → 25 like one, 5 neither

Watch out

Reading OR as “only one”: in logic, P OR Q is also True when both are True.
Applying NOT to only part of a bracket, or forgetting the brackets.
Adding the two groups without taking away the overlap.

Worked example

(a) Rule: if it is a Saturday or a Sunday, the library opens at 10 a.m.; on other days it opens before 10 a.m.
Thursday: opens at 10 a.m.
Tuesday: opens at 8 a.m.
Friday: opens at 9 a.m.
Saturday: opens at 10 a.m.
Sunday: opens at 10 a.m.
Which day breaks the rule?
(b) P is False and Q is True. Is "NOT P OR NOT Q" True or False?

Working: (a) Thursday is a weekday but opens at 10; (b) put P = False, Q = True and work out the brackets first

Answer: (a) Thursday; (b) True