**PROPOSITIONAL LOGIC**

A

**proposition**is a declarative sentence (that is, a sentence that declares a fact) that is either true or false, but not both.**a.**The

**of p, denoted by ¬p (also denoted by p), is the statement “It is not the case that p.” The proposition ¬p is read “not p.” The truth value of the negation of p, ¬p, is the opposite of the truth value of p.**

*negation*

**b.**The

**of p and q, denoted by p ∧ q, is the proposition “p and q.” The conjunction p ∧ q is true when both p and q are true and is false otherwise.**

*conjunction*

**c.**The

**of p and q, denoted by p ∨ q, is the proposition “p or q.” The disjunction p ∨ q is false when both p and q are false and is true otherwise.**

*disjunction*

**d.**The

*XOR (*

*exclusive***of p and q, denoted by p ⊕ q, is the proposition that is true when exactly one of p and q is true and is false otherwise.**

*or)*

e.
The

**of p → q is the proposition “if p, then q.” The conditional statement p → q is false when p is true and q is false, and true otherwise.***implication*
f.
The

**statement p ↔ q is the proposition “p if and only if q.” The statement p ↔ q is true when p and q have the same truth values, and is false otherwise.***bi-implications*

*Logic and Proofs*

*Discrete Mathematics**“*

**Resume”**fropky.com |

**MATHEMATICAL PROOFS**

A proof is a
valid argument that establishes the truth of a mathematical statement. A proof
can use the hypotheses of the theorem, if any, axioms assumed to be true, and
previously proven theorems.

**a.**

**Direct Proofs.**A direct proof of a conditional statement p → q is constructed when the first step is the assumption that p is true; subsequent steps are constructed using rules of inference, with the final step showing that q must also be true. A direct proof shows that a conditional statement p → q is true by showing that if p is true, then q must also be true, so that the combination p true and q false never occurs.

**b.**

**Proofs by Contradiction.**Another type of

**proof. Suppose we want to prove that a statement p is true. The contradiction q such that ¬p → q is true. Because q is false, but ¬p → q is true, we can conclude that ¬p is false, which means that p is true. Because the statement r ∧¬r is a contradiction whenever r is a proposition, we can prove that p is true if we can show that ¬p → (r ∧¬r) is true for some proposition r.**

*indirect*

**c.**

**Proof by Contraposition.**Another type of

**proof. Use of the fact that the conditional statement p → q is equivalent to its contrapositive, ¬q →¬p. This means that the conditional statement p → q can be proved by showing that its contrapositive, ¬q →¬p, is true. In a proof by contraposition of p → q, we take ¬q**

*indirect***as a premise, and**

**using axioms, definitions, and previously proven theorems, together with rules of inference, we show that ¬p must follow.**

*Reference: Discrete_Mathematics_and_Its_Applications_7th_Edition_Rosen*

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