MM250 DISCRETE MATH LAB Command Center

1. Logical Reasoning

VERIFIED AGAINST MM250 MATERIAL

Sections: 1.1 · 1.2 · 1.3 · 1.4 · 1.5 · 1.6

EVALUATORVERIFIED AGAINST MM250 MATERIAL

Proposition Evaluator

§1.1–1.3 · Evaluate a compound proposition for a chosen truth assignment.

REMEMBER: OR (∨) normally means inclusive OR — false only when both sides are False.
zyBooks-confirmed precedence: ¬ > ∧ > ∨, and ¬/∧/∨ always evaluate before → or ↔. zyBooks does not order → against ↔, or ⊕ against anything — mixing either of those pairs without explicit parentheses is rejected rather than guessed. Chained ⊕⊕ is accepted as an extra capability beyond the course, since zyBooks doesn't address chaining XOR at all.
VISUALIZERVERIFIED AGAINST MM250 MATERIAL

Truth Table Generator

§1.1–1.4 · Build the complete truth table for an expression and classify it.

zyBooks-confirmed precedence: ¬ > ∧ > ∨, evaluated before → or ↔. → vs ↔ and ⊕ vs anything are not ordered by the course — mixing either pair without explicit parentheses is rejected rather than guessed.
STEPPERVERIFIED AGAINST MM250 MATERIAL

Reverse Truth-Table Helper

§1.4–1.6 · Build an expression from a desired True/False output column.

REMEMBER: different-looking expressions may still be logically equivalent — this method gives one valid answer, not the only one.
EVALUATORVERIFIED AGAINST MM250 MATERIAL

De Morgan / Equivalence Lab

§1.4 · Compare two expressions and determine whether they are logically equivalent.

REMEMBER: De Morgan pushes the negation inward AND changes the connective (∧ ↔ ∨).
zyBooks-confirmed precedence: ¬ > ∧ > ∨, evaluated before → or ↔. → vs ↔ and ⊕ vs anything are not ordered by the course — mixing either pair without explicit parentheses is rejected rather than guessed.
REFERENCEVERIFIED AGAINST MM250 MATERIAL

Conditional / Biconditional Quick Tool

§1.3 · See at a glance when p → q and p ↔ q are True or False.

REMEMBER: p → q is False only when p is True and q is False — "the promise is made and broken." p ↔ q is True whenever the two truth values match.
REFERENCEVERIFIED AGAINST MM250 MATERIAL

Proposition & Symbol Reference

§1.1 · Core vocabulary and notation for propositional logic.

Proposition
A declarative statement that is either True or False, but not both.
Simple proposition
A single proposition with no logical connectives — one basic statement.
Compound proposition
Two or more simple propositions joined by a logical connective (¬, ∧, ∨, ⊕, →, ↔).
Truth value
Whether a proposition is True or False.
Symbols
p, q, r, s are used to represent propositions.
0 / 1 shorthand
0 = False, 1 = True.
REFERENCEVERIFIED AGAINST MM250 MATERIAL

Conditional Language Helper

§1.3 · Different ways the same conditional p → q gets phrased in English.

All of the following mean exactly p → q:

  • If p, then q
  • If p, q
  • q if p
  • p implies q
  • p only if q
  • p is sufficient for q
  • q is necessary for p
REMEMBER: "p only if q" and "q is necessary for p" are the two students mix up most — both still mean p → q, not q → p.
EVALUATORVERIFIED AGAINST MM250 MATERIAL

Converse / Contrapositive / Inverse Helper

§1.3 · See all four related conditionals for p → q side by side, with their truth values.

REMEMBER: a conditional and its contrapositive always share the same truth value. The converse and inverse also always match each other — but neither necessarily matches the original.
EVALUATORVERIFIED AGAINST MM250 MATERIAL

Logical Argument Validity

§1.5 · Test whether a conclusion follows from a set of hypotheses.

REMEMBER: an argument is VALID only if the conclusion is True in every row where all hypotheses are True — one counterexample row makes it invalid.
STEPPERVERIFIED AGAINST MM250 MATERIAL

Boolean Algebra Lab

§1.6 · Evaluate and step-by-step simplify Boolean expressions, naming the law used at each step.

Use + for OR, · or * (or just write letters together, like xy) for AND, and bar(...) or a trailing ' for complement — e.g. bar(x+y), x', x.x'+y.1

Idempotent
x + x = x    x·x = x
Associative
(x + y) + z = x + (y + z)    (xy)z = x(yz)
Commutative
x + y = y + x    xy = yx
Distributive
x + yz = (x + y)(x + z)    x(y + z) = xy + xz
Identity
x + 0 = x    x·1 = x
Domination
x + 1 = 1    x·0 = 0
Double complement
bar(bar(x)) = x
Complement
x + bar(x) = 1    x·bar(x) = 0    bar(0) = 1    bar(1) = 0
De Morgan
bar(x + y) = bar(x)·bar(y)    bar(xy) = bar(x) + bar(y)
Absorption
x + xy = x    x(x + y) = x
The step-by-step simplifier automatically applies Idempotent, Identity, Domination, Complement, Double Complement, Absorption, and De Morgan (as a last resort to unlock further reduction). Associative, Commutative, and Distributive are listed above for reference — they don't have one single "simplifying" direction, so they aren't auto-applied.
EVALUATORVERIFIED AGAINST MM250 MATERIAL

Boolean Equivalence Checker

§1.6 · Compare two Boolean expressions for equivalence (course notation uses =, not ≡).