Truth Tables

negation of p¬p p ~p T F F T

ENGLISH EXPRESSION: "not p" "it is not the case that p"

Rule for Negation Negation switches the truth value. Negation of p Truth value: True if and only if p is false Rule for Conjunction A conjunction is true when both parts are true conjunction ^ and doublr true = true Rule for Disjunction A disjunction is false when both parts are false. disjunction V or double false = false Rule for the Conditional A conditional is false for if true then false.conditional p-true, q-false = false Rule for the Biconditional A biconditional is true when the parts have the same truth values. bi-conditional double true or double false = true

Negation ¬p [not p] True when P is false. Conjunction p^q [p and q] True when both p and q are true. Disjunction p\/q [p or q] True when either p or q, or both p and q are true. Exclusive Disjunction p\_/q [p or q but not both] True when either p or q is true but not both. Implication p=> [If...then] True unless p is true and q is false. Equivalence p<=>q [If and only if] True when both p and q are true or both p and q are false.

biconditional p ↔ q p q T T T T F T F T T F F F

Truth Value Only true when p and q are both true or when p and q are both false. ENGLISH EXPRESSION: "p if and only if q" "p is necessary and sufficient for q" "if p, then q, and conversely"

disjunction (v) or p∨q pq T T T T F T F T T F F F

Truth Value: Only false when both p and q are false. ENGLISH EXPRESSION: "p or q (or both)" "p unless q"

conditional p → q p q T T T T F F F T T F F F

Truth value: Only false when p is true and q is false. ENGLISH EXPRESSION: "if p, then q" "p implies q" "a necessary condition for p is q"

conjunction p∧q p q T T T T F F F T F F F F

Truth value: Only true when both p and q are true. ENGLISH EXPRESSION: "p and q" "p but q"

~ opposite value ^ true only when both parts are true v true when at least one part is true -> False only when true hypothesis leads to false conclusion <-> true when p and q are both true or both false chain rule p->q q->r ----- p->r disjunctive inference p v q or p v q ~p q->r ---- ----- q p

Conditional If (A) then (B) Converse If (B) then (A) Inverse If Not(A) then not (B) Contrapositive If Not(B) then Not (A) Biconditional Iff(A) then (B) Union Both A And B must be true INtersection A or B should be true Conjunction Truth Table ( __r_ • _t__ ) and ^ Disjunction Truth Table ( r v p ), Or v Biconditional Truth Table ( b<-> s ) (triple bar)iff Negation Truth Table ~p Conditional Truth Table ( P⊃ Q ) P->Q if P, then Q

Conditional If p then q p→q Converse If q then p q→p Inverse If ∼p then ∼q ∼p→∼q Contrapositive If ∼q then ∼p ∼q→∼p

Conjunction Truth Table ( __r_ • _t__ ) and ^ Disjunction Truth Table ( r v p ), Or v Biconditional Truth Table ( b<-> s ) (triple bar)iff Negation Truth Table ~p Conditional Truth Table ( P⊃ Q ) P->Q if P, then Q

1 member universe: (x)Px Pa 1 member universe: (∃x)Px Pa 2 member universe: (x)Px Pa • Pb 2 member universe: (∃x)Px Pa ∨ Pb 2 member universe: (x)(Px ⊃ Qx) (Pa ⊃ Qa) • (Pb ⊃ Qb) 2 member universe: (x) Px ⊃ (∃x)Qx (Pa • Pb ) ⊃ (Qa ∨ Qb) 3 member universe: (x)Px Pa • Pb • Pc 3 member universe: (∃x)Px Pa ∨ Pb ∨ Pc 3 member universe: (x)(Px ⊃ Qx) (Pa ⊃ Qa) • (Pb ⊃ Qb) • (Pc ⊃ Qc) 3 member universe: (x)Px ⊃ (∃x)Qx (Pa • Pb • Pc ) ⊃ (Qa ∨ Qb ∨ Qc) ~{[(A·B)·(~X·Z)]·~[A·B)v~(~Y·~Z)]} ~{[(T·T)·(~F·F)]·~[T·~T)v~(~F·~F)]} ~{[(F·T)·(T·F)]·~[T·F)v~(~T·~T)]} ~{[F·F)·~(FvF)]} ~{F·~F} ~{F·T} ~{F} T [Av(BvC)]·~[(AvB)vC] [Tv(TvT)]·~[(TvT)vT] [TvT)]·~[(TvT] T·~T F·F F ~AvB ~TvT FvT T (B·C)v(Y·Z) (T·T)v(F·F) TvF F

P ∼P T ? F ? ∼P F T P Q P&Q T T ? T F ? F T ? F F ? P&Q T F F F P Q P∪Q T T ? T F ? F T ? F F ? P∪Q T T T F P Q P⊃Q T T ? T F ? F T ? F F ? P⊃Q T F T T P Q P≡Q T T ? T F ? F T ? F F ? P≡Q T F F T Please allow access to your computer's microphone to use Voice Recording. Having trouble? Click here for help.

Converse (Implications) Q implies P Inverse (Implications) not P implies not Q Contrapositive (Implications) not Q implies not P Law of Detachment If P, then Q P is true therefore Q is true Law of Excluded Middle A mathematical statement ,P, is either true or false Law of Contradition No statement is both true and false * The negation of a True statement is False as is a false statement is true. Conditional Statement Formed by joining two statements, P and Q, with the words, if and then.

X ∩ X = X X ∪ Y = Y ∪ X (X ∪ Y)′ = X′ ∪ Y′ X ∩ (Y ∩ Z) = (X ∩ Y) ∩ Z ... (X ∩ Y) ∪ Z = (X ∪ Z) ∩ (Y ∪ Z) ... ∅ ∩ X = ∅ I ∪ X = I X ∪ X′ = I X ∩ X′ = ∅ X ∪ Y = Y X ∪ (Y ∩ Z) = (X ∪ Y) ∩ Z ... (X ∩ Y)′ = X′ ∪ Y′ (X ∪ Y)′ = X′ ∩ Y′ ... X ∩ (X ∪ Y) = X (X′ ∪ Y′)′ ∪ (X′ ∪ Y)′ = X ... X ∪ (X ∩ Y) = X X ∪ X = X X ∩ Y = Y ∩ X X ∪ (Y ∪ Z) = (X ∪ Y) ∪ Z ... (X ∪ Y) ∩ Z = (X ∩ Z) ∪ (Y ∩ Z) ... ∅ ∪ X = X I ∩ X = X (X′)′ = X X ∩ Y = X⇔X ⊂ Y⇔X ∪ Y = Y ... X ⊂ X ... X ⊂ Y and Y ⊂ Z ⇒ X ⊂ Z ... ∅ ⊂ X ... X ⊂ I ... A ∩ X = A ∩ Y and A ∪ X = A ∪ Y⇒ X = Y ... X ⊂ X ∪ Y ... X ∩ Y ⊂ X ... T = (X ∩ T′) ∪ (X′ ∩ T)⇔X = ∅ ... X ⊂ A′ ⇒ X ∩ A = ∅ ... A′ ⊂ Y⇔A ∪ Y = I ... X ∩ Y′ = Z ∩ Z′⇔X ∩ Y = X ... X ⊂ Y⇔Y′ ⊂ X′ ... X ⊂ Y and X ⊂ Z ⇒ X ⊂ Y ∩ Z ... X ⊂ Z and Y ⊂ Z ⇒ X ∪ Y ⊂ Z .. X = Y⇔X ⊂ Y and Y ⊂ X ... X ⊂ Y⇔X′ ∪ Y = I ... X ⊂ Y⇒ X ∪ Z ⊂ Y ∪ Z ... X ⊂ Y⇒ X ∩ Z ⊂ Y ∩ Z ... ∅′ = I I′ = ∅

Set theory Union Given two sets, it is the set of elements that are members of either set. The symbol is U. "Joined Together" Intersection Given two sets, it is the set of elements that are members of both sets. "What's in Common" Complement The elements within the universal set which are not contained in a given set. "what's left" or "bench-warmers" Roster Notation Elements of a set are listed between braces, { } Ex: { 1, 2, 3, 4, 5, 6, 7, ..} Interval Notation a way of writing the set of all real numbers between two endpoints. the symbols [and] are used to include an endpoint in an interval, and the symbols (and) are used to exclude endpoint from an interval. Ex: [2, 3) includes all numbers greater than or equal to 2 but less than 3. Set-builder Notation uses the properties of the numbers in the set to define the set Ex: {x l x∈R, 0< x <1}

¬(A ∪ B) = ¬A ∩ ¬B ¬(A ∩ B) = ¬A ∪ ¬B DeMorgan's Law A ∪ B = B ∪ A A ∩ B = B ∩ A Commutative Law A ∪ (B ∪ C) = (A ∪ B) ∪ C A ∩ (B ∩ C) = (A ∩ B) ∩ C Associative Law A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C) A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C) Distributive Law A U B [A union B] Everything that is in either or both sets A n B [A intersection B] Everything that is in the overlap of both sets A' [The complement of A] Everything that is not in set A (A U B)' [The compliment of (A U B) Everything that is not in set A or set B A n B' [The intersection of A and the intersection of B] The overlap between A and everything that is not in B A' U B [The union of B with the Complement of A] Everything that is in B and not in A (A n B)' [The compliment of A intersection B] Everything that is not in the overlap of A and B