Propositions and Truth Tables

• Study notes on Propositions and Truth Tables

Propositional Logic Operators

• Negation uses the symbol ¬ and means "not"
• Conjunction uses the symbol ∧ and means "and"
• Disjunction uses the symbol V and means "or"
• Implication uses the symbol → and means "if...then"
• Equivalence uses the symbol ↔ and means "if and only if"

Converting Statements to Symbols

• p = "I love discrete math."
• q = "I like solving math problems."
• r = "I am an IT student"

Converting the following symbols to statements:

• ¬p means "I do not love discrete math" or "It is not true that I love discrete math."
• p ∧ q means "I love discrete math and I like solving math problems."
• p v q means "I love discrete math or I like solving math problems."
• p → q means "If I love discrete math, then I like solving math problems."
• ¬(p^q) means "It is not true that I love discrete math and I like solving math problems," or "I do not love discrete math or I do not like solving math problems."

Converting Statements to Symbols

• p = "I love discrete math"
• q = "I like solving math problems"
• r = "I am an IT student"

Converting the following statements to symbols:

• "I don't like discrete math or I like solving math problems" translates to ¬p v q
• "If I am an IT student, then I don't love discrete mathematics" translates to r → ¬p
• "I love discrete math or I like solving math problems" translates to p v q
• "It is not the case that I like discrete math or I don't like solving math problems" translates to ¬(p v ¬q)
• "If I don't like discrete math, then I don't like solving math problems" translates to ¬p → q

Answers to the Question:

• ¬p → ¬r means "If I do not like sports, then I am not a sports athlete."
• (¬p ^ ¬q) → ¬r means "If I do not like sports and I do not love playing ball games, then I am not a sports athlete."
• (p^q) → r means "If I like sports and I love playing ball games, then I am a sports athlete."
• r ↔ (p^q) means "I am a sports athlete if and only if I like sports and I love playing ball games."

Tautologies, Contradiction, and Contingencies

• Tautology is a proposition that is always true
• Contradiction is a proposition that is always false.
• Contingency is a proposition that is sometimes true and sometimes false.
• Truth Table Construction: Number of rows = 2^n (where n is the number of variables)

Truth Table Construction Rules

• Determine the number of rows (2^n).
• List the independent variables.
• Evaluate negations (¬).
• Evaluate expressions within parentheses.
• Evaluate expressions within square brackets.
• Evaluate the whole expression.

Logical Equivalence

• Two propositions, p and q, are logically equivalent (p ≡ q) if p ↔ q is a tautology
• Example: ¬p v q = p → q
• T (or 1) = Tautology (always true)
• F (or 0) = Contradiction (always false)

Propositional Logic Laws

• p^T = p (and True is the proposition itself)
• "The sun is shining" AND "True" is equivalent to "The sun is shining”
• pv F = p (or False is the proposition itself)
• "The cat is black" OR "False" is equivalent to "The cat is black."
• Domination Laws:
• pv T = T (or True is always True)
• Example: "The sky is blue" OR "True" is always "True."
• p^ F = F (and False is always False)
• Example: "The car is red" AND "False" is always "False."

Double Negation Law:

• ¬¬p = p (not not p is p)
• Example: "It is not not raining" is equivalent to "It is raining."

De Morgan's Laws:

• ¬(p^q) = ¬p v ¬q (not (p and q) is not p or not q)
• Example: "It is not (hot and humid)" is equivalent to "It is not hot or it is not humid."
• ¬(pv q) = ¬p ^ ¬q (not (p or q) is not p and not q)
• Example: "It is not (sunny or cloudy)" is equivalent to "It is not sunny and it is not cloudy.”

Distributive Laws:

• p^ (qvr) = (p^q) v (p^ r) (and distributes over or)
• Example: "I will eat (apples or oranges) and pears" is equivalent to "(I will eat apples and pears) or (I will eat oranges and pears)."
• pv (q^r) = (pv q) ^ (pv r) (or distributes over and)
• Example: "I will eat apples or (oranges and pears)" is equivalent to "(I will eat apples or oranges) and (I will eat apples or pears)."

Absorption Laws:

• p^ (pvq) = p
• Example: "I will eat apples and (apples or bananas)" is equivalent to "I will eat apples."
• pv (p^q) = p
• Example: "I will eat apples or (apples and bananas)" is equivalent to "I will eat apples."

Commutative Laws:

• p^q=q^p (order doesn't matter for and)
• pvq=qvp (order doesn't matter for or)
• Associative Laws:
• p^ (q^r) = (p^q) ^ r (grouping doesn't matter for and)
• pv (qvr) = (p v q) v r (grouping doesn't matter for or)

Inverse Laws:

• p^ ¬p = F (p and not p is always false)
• pv ¬p = T (p or not p is always true)

Conditional Law:

• p→ q = ¬pv q (if p then q is not p or q)