Propositions, connectives and quantifiers
True-or-false statements, and / or / not / if-then, truth tables, negating a statement with "all" or "some".
First, the rules
Statements that are true or false
A proposition is a sentence that is definitely true or definitely false — "7 is prime" is one, "close the door" is not. Logic is the arithmetic of such sentences.
| p | q | p ∧ q | p ∨ q | p → q | p ↔ q |
|---|---|---|---|---|---|
| T | T | T | T | T | T |
| T | F | F | T | F | F |
| F | T | F | T | T | F |
| F | F | F | F | T | T |
"If it rains I stay home" is contradicted by: it rains AND I do not stay home.
See it
Flip p and q, light the lamp
Two switches, one statement. Flip p and q and watch the lamp; the table row that matches lights up. Pick a second statement to compare columns — identical columns mean the two are equivalent.
Step by step
Worked examples
Building a truth table
connectivesMake the truth table of (p → q) ∧ ¬q and decide for which values it is true.
Ask yourself: Work column by column: first p → q, then ¬q, then the ∧ of those two.
Show the solution
- Rows TT, TF, FT, FF. p → q gives T, F, T, T. ¬q gives F, T, F, T.
- The ∧ is true only where both columns are T: only the last row (p false, q false).
- So the statement is true exactly when p and q are both false.
Answer true only for p = F, q = F
Converse, inverse, contrapositive
implicationsStatement: "If a number ends in 0, then it is divisible by 5." Write the three related forms and say which are true.
Ask yourself: The original is true. Which rearrangement must have the same truth value?
Show the solution
- Converse: "If a number is divisible by 5, then it ends in 0." — false (15 is divisible by 5).
- Inverse: "If a number does not end in 0, then it is not divisible by 5." — false (same counterexample, 15).
- Contrapositive: "If a number is not divisible by 5, then it does not end in 0." — true, like the original; it is the same statement in disguise.
Answer contrapositive true; converse and inverse false
Negating with care
De Morgan · quantifiersNegate: (a) "Every student in the class is over 16." (b) "x is even and x is positive." (c) "If I study, I pass."
Ask yourself: "Every" becomes "some … not"; "and" becomes "or" with each part negated; an implication is denied by the "if" being true and the "then" failing.
Show the solution
- (a) "Some student in the class is not over 16" — one counterexample is enough; "no student is over 16" says far too much.
- (b) "x is odd or x is not positive" (De Morgan: negate each part, swap ∧ for ∨).
- (c) "I study and I do not pass."
Answer (a) some … not · (b) odd OR not positive · (c) study AND not pass
Practice: logic
0 / 0Negate a statement, evaluate a compound statement, name the converse / inverse / contrapositive, or count the rows of a truth table. Your best score is kept in this browser only.
About this lesson
Propositions, connectives and quantifiers: true-or-false statements, and / or / not / if-then, truth tables, the converse, inverse and contrapositive of an implication, De Morgan’s laws, and how to negate a statement with “all” or “some”. Worked examples and practice questions with explanations.
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