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Tautology

A statement that is ALWAYS TRUE regardless of the truth values of its components. Example: p ∨ ¬p

Contradiction

A statement that is ALWAYS FALSE regardless of the truth values of its components. Example: p ∧ ¬p

Contingency

A statement that is sometimes true and sometimes false (final column has both T and F). Examples: p → q, p ∧ q

"The client is authenticated OR the client is not authenticated." Classify it.

Tautology (form: p ∨ ¬p, always true). Same for "Either the webpage is loading or the webpage is not loading."

"The router is connected AND the router is not connected." Classify it.

Contradiction (form: p ∧ ¬p, always false)

"If the server responds, then the website is accessible." Classify it.

Contingency (it is a conditional p → q, true in some rows and false in others). NOT a tautology. Shade the answer, not E!

"If the IP address is valid, then the device can connect to the network." Classify it.

Contingency (conditional p → q; false only when p is T and q is F)

"If the password is correct, then the user can log in." Classify it.

Conditional (if-then, p → q)

"You can graduate if and only if you have completed all required courses." Classify it.

Biconditional (p ↔ q)

Conditional p → q: when is it FALSE?

Only when p is TRUE and q is FALSE. True in all other cases.

Biconditional p ↔ q: when is it TRUE?

When p and q have the SAME truth value (both T or both F). False when they are opposite.

Biconditional p ↔ q can be broken down into:

Two conditionals combined with AND: (p → q) ∧ (q → p)

Which best describes a biconditional statement?

It is true only when BOTH conditionals (p → q and q → p) are true.

Relationship between conditional and biconditional

p ↔ q ≡ (p → q) ∧ (q → p)

"If the system is secure, then the firewall is active AND if the firewall is active, then the system is secure." Equivalent to:

Biconditional: (p → q) ∧ (q → p) ≡ p ↔ q

Key words that signal a biconditional

"if and only if", "iff", or two conditionals going both directions (if p then q AND if q then p)

Conditional rewritten without an arrow

p → q ≡ ¬p ∨ q (p → q and ¬p ∨ q are logically equivalent)

How many rows does a truth table have?

2ⁿ rows, where n = number of DIFFERENT variables. (p ∨ q) ∧ r has 3 variables → 8 rows (not 4, 5 or 6). Watch out: the answer may be 'E: none of the above' if 8 is not listed!

Standard row order for p, q, r (8 rows)

TTT, TTF, TFT, TFF, FTT, FTF, FFT, FFF (p: 4 T then 4 F; q: 2 T, 2 F repeating; r alternating T, F)

Steps to build a truth table for a compound proposition

1) List all T/F combos of the variables. 2) Make a column for each small part (negations, inner parentheses first). 3) Combine step by step. 4) The last column is the answer for the whole statement.

Logical equivalence (≡)

Two statements are logically equivalent if they have IDENTICAL truth values in every row (their final columns match exactly).

How do you prove logical equivalence with a truth table?

Build columns for both sides, then show the two final columns are identical in all rows. Write a conclusion: "Both columns are the same, therefore the statements are logically equivalent."

Truth table 1: [(p → q) ∧ (q → r)] ∨ ¬p — final column (rows TTT to FFF)

T, F, F, F, T, T, T, T (contingency). Columns: p→q, q→r, (p→q)∧(q→r), ¬p, then final.

Truth table 2: p ∨ (q ∧ r) ≡ (p ∨ q) ∧ (p ∨ r) — final column

Both sides: T, T, T, T, T, F, F, F → identical, so equivalent (Distributive Law).

Distributive Law of logic

p ∨ (q ∧ r) ≡ (p ∨ q) ∧ (p ∨ r) and p ∧ (q ∨ r) ≡ (p ∧ q) ∨ (p ∧ r)

(p → q) ∧ (q → r): which rows are true? (rows TTT to FFF)

T, F, F, F, T, F, T, T. It is the hypothesis of Hypothetical Syllogism.

Proposition

A declarative sentence that is either TRUE or FALSE, but not both.

Which of these is a proposition? "Close the door." / "Is it raining?" / "2 + 3 = 5" / "x + 1"

"2 + 3 = 5" only. Commands, questions, and expressions with an unknown variable are NOT propositions.

Atomic (simple) proposition

A single statement with no connectives. Example: "The printer is online.", "The Earth revolves around the Sun.", "The student passed the exam."

Compound proposition

Two or more propositions joined by logical connectives. Example: "The server is running AND the database is accessible." (p ∧ q)

Logical connectives and symbols

Negation ¬ (not), Conjunction ∧ (and), Disjunction ∨ (or), Conditional → (if…then), Biconditional ↔ (if and only if)

Conjunction p ∧ q: when true?

Only when BOTH p and q are true.

Disjunction p ∨ q: when true?

When at least one of p, q is true. False only when both are false.

Negation ¬p

Reverses the truth value: ¬T = F, ¬F = T.

Are p → q and ¬p ∨ q equivalent?

YES. Both are false only when p = T and q = F.

Are p ∨ q and p ∧ q equivalent?

NO. They differ when p and q have opposite truth values.

Are p ↔ q and p ∨ q equivalent?

NO. Example: p = T, q = F gives p ∨ q = T but p ↔ q = F.

Are ¬(p ∨ q) and ¬p ∨ ¬q equivalent?

NO. De Morgan: ¬(p ∨ q) ≡ ¬p ∧ ¬q (AND, not OR).

De Morgan's Laws

¬(p ∧ q) ≡ ¬p ∨ ¬q and ¬(p ∨ q) ≡ ¬p ∧ ¬q

Converse, inverse, contrapositive of p → q

Converse: q → p. Inverse: ¬p → ¬q. Contrapositive: ¬q → ¬p. Only the contrapositive is equivalent to p → q.

Quick classification trick

Look at the FORM: p ∨ ¬p → tautology; p ∧ ¬p → contradiction; plain if-then or and/or of different statements → contingency (unless a truth table shows otherwise).

Exam format reminder (Enabling Assessment 1)

20 multiple choice (shade E if none of the choices is correct) + 2 truth tables (5 pts each, list all T/F values of p, q, r).

Exam tip: when to shade E

ONLY if NONE of A to D is correct. Before choosing E, re-check the options. Missed items last time: #13, #15, #16, #20 (all had a correct choice A to D).