Logical Equivalences

Lecture 3, DSCI 220, 2026W1

Announcements

• Tutorials begin this week.

  • L1D    Tue 2–4    ORCH 3002
  • L1B    Tue 4–6    ORCH 3002
  • L1A    Wed 1–3    SPPH 143

• No class this Friday. Video released by Thursday morning.

• HW1 available, due Sunday, Sep 20.

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Propositions Continued

Parse Trees

<wff> ::= <atom>
        | ~ <wff>
        | ( <wff> ∧ <wff> )
        | ( <wff> ∨ <wff> )

<atom> ::= p | q 

Apply the outermost relevant rule…

\(((\text{Tu} \wedge \text{Th}) \wedge (\text{Sa} \vee \text{Su}))\)

\((\text{Tu} \wedge (\text{Th} \wedge (\text{Sa} \vee \text{Su})))\)

Work outward-in: the outermost parentheses are the last rule applied, so they are the root. Each internal node is a rule application, each leaf an atom.

EXPECT THE QUESTION: “how do we know which operator goes with the outer parentheses?” Answer by annotating, not by algorithm – say outright that the programmatic version waits for 221, and that for now this is observational.

Write the depth under each operator:

  ((Tu ^ Th) ^ (Sa v Su))          (Tu ^ (Th ^ (Sa v Su)))
      2       1     2                  1      2      3

Depths 2,1,2 – the middle one is the root. Depths 1,2,3 – the first one is. Same symbols, different tree, visible without parsing anything.

Counting version, if they want to do it by hand: the root is the operator with exactly ONE more “(” than “)” to its left. Same fact, two tallies instead of a running counter.

Why it works is the grammar: ( op ) wraps every binary operator in its OWN parentheses, so an operator’s depth counts how many rule applications deep it sits. Exactly one sits at depth 1.

Edge case, and someone will write it: ~ adds no parentheses, so on ~(p ^ q) the counting rule points at the ^ when the root is the leading ~. Rule of thumb – if it starts with ~, that is the root; otherwise count.

The trees are different – and every row of the truth table agrees. That gap is the point. The parentheses fix the structure, the structure fixes how you evaluate, but two structures can evaluate the same everywhere. What the parentheses buy you is a definite answer, not a unique expression.

That is the opening for Monday: logical equivalence is exactly this relation – different formulas, same truth table. Do not name it yet; just let them notice that two different trees came out the same.

WFFs

<wff> ::= <atom>
        | ~ <wff>
        | ( <wff> ∧ <wff> )
        | ( <wff> ∨ <wff> )

<atom> ::= p | q 

Theorem: \(\neg (p ∨ q))\) is not a WFF.

Proof:

Back to Propositions

A proposition is a statement that can be either True or False.

Examples:

• \(37 > 12\)

• Fewer than 5 people in this room feel sleepy.

• There are extra-terrestrial life forms.

• This statement is False.

• \(\underline{\hspace{10em}}\)

Translation

From English to logic and back…

\(p\): ___________ ate cereal for breakfast.

\(q\): ___________ brought a backpack to class.

 

• \((p\lor q)\)

• \((p\wedge q)\)

• \(\neg q\)

Logical Equivalence

Logical Equivalence

Ex: Is it true that \((p \lor q) \equiv (q\lor p)\) ?

Ex: Is it true that \(((\text{Tu} \wedge \text{Th}) \wedge (\text{Sa} \vee \text{Su})) \equiv (\text{Tu} \wedge (\text{Th} \wedge (\text{Sa} \vee \text{Su})))\) ?

Discussion points:

• \(\equiv\) means logically equivalent

• The answer had better be _________!!!

• How can we justify our instinct?

1. Show same truth table
2. Chains of equivalence

Same as saying “write a proof”

The second example is the swim pair from the parse-tree slide – two different trees, and they noticed the truth tables agreed. This is the name for what they saw. The first example they will believe instantly; the second is the one that earned the question.

Logical Equivalences

We have discussed Logical Equivalence (LE) and explored an example. The table illustrates LE that are so important, they have names!

Eight cells are missing. What belongs in them?

Observations:

The blanks are in Identity, Domination and Negation. Identity and Negation hide two cells each; Domination hides four, because both the operand and the result are greyed on each of its lines. All eight are T or F, and all eight follow from the \(\wedge\) and \(\vee\) truth tables – no new vocabulary needed.

Spend the last bullet on Negation: \(p \vee \neg p \equiv\) T and \(p \wedge \neg p \equiv\) F are tautology and contradiction. Filling those two cells is the next slide; it only supplies the names.

De Morgan is two rows further down – flag it, since the cereal slide verifies it.

Special Logical Equivalences

Suppose \(p\equiv\) T . Then \(p\) is a Tautology.

Suppose \(p\equiv\) F . Then \(p\) is a Contradiction.

Logical Equivalence

Suppose we have the following propositions, and someone has asserted that for Cheerios, \(\neg(p\lor q)\). How is this expressed in English? Can you find a simpler expression?

\(p\): a cereal has less than 4 units of protein

\(q\): a cereal has more than 6 units of sugars

DeMorgan’s tells us \(\neg p\wedge \neg q \equiv \neg( p\lor q)\).

\(p\)	\(q\)	\(\neg p\wedge \neg q\)
F	F
F	T
T	F
T	T

\(p\)	\(q\)	\(\neg( p\lor q)\)
F	F
F	T
T	F
T	T

Simplified statement:

half the class complete the TT on the left, the other half on the right, then we all read our responses aloud!

Chains of Equivalence

Prove that

\[(((p \wedge q) \wedge r) \vee ((p \wedge q) \wedge \neg r)) \;\equiv\; (p \wedge q)\]

\(\equiv\)

\(\equiv\)

\(\equiv\)

The other method. Truth tables settled the last one; this settles it by rewriting, one named law at a time.

Three variables, so a truth table is 8 rows. The chain is 3 lines. Say that out loud – it is the whole argument for learning the laws.

(((p ^ q) ^ r) v ((p ^ q) ^ ~r)) == ((p ^ q) ^ (r v ~r)) Distributive == ((p ^ q) ^ T) Negation == (p ^ q) Identity

Every line is a WFF – fully parenthesised, one binary operator per pair. Worth being fussy about here, one slide after parse trees.

r vanishes. That is the point to sit on: r appears twice in the original and cannot change the answer. They meet exactly this again in HW1.

This is also the thing Tutorial 1 asks for tomorrow, and this is the only worked example they get before it.

Logical Equivalence Practice

Navigate to Activity 3 and complete question 1. Each of you will receive a different version of the problem. Work together in pairs to solve each person’s version.

PrairieLearn Activity 3

Words and Ideas

Grammar

Logical Equivalence

Tautology, Contradiction