Implication in Mathematics, Summaries of Mathematics

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Simple Implications

One of the examples we had previously proved as tautology is an example of a Simple Implication.

This Simple Implication is called Modus Ponens

If a car has a dead battery, then the car will not start. Jim’s car has a dead battery.

Example

Conclusion: Jim’s car will not start.

If two lines are parallel, then they do not

intersect. Line f is parallel to line m.

Conclusion: Lines f & m do not intersect.

Example

f

m

If you spend time with friends, then you enjoy yourself. If you enjoy yourself, then your time is well spent.

Example

Conclusion: If you spend time with friends, then your time is well spent.

If two planes are not parallel, then they intersect. If two planes intersect, then they intersect in a line.

If two planes are not parallel, then they intersect in a line.

Example

Simple Implications

are tautologies that are very useful and

often used.

Other Simple Implications

Modus Tollens – the process of denying the

consequent

[(p → q) ∧ ~ p+ → ~ p

Simplification

(p ∧ q) → p

Equivalence Implications

are also tautologies often used to

validate statements

De Morgan’s Law

the negation of a disjunction is the conjunction

of the negations; and the negation of a

conjunction is the disjunction of the negations;

∼(p∧q)↔(∼p∨∼q)

∼(p∨q)↔(∼p∧∼q)

Other Equivalence Implications

Idempotent Law

(p ∧ p) ↔ p (p ∨ p) ↔ p

Double Negation

p ↔ ~ (~ p)

Contraposition

(p → q) ↔ (~ q →~ p)

Other Equivalence Implications

Equivalence for Implication and Disfunction

(p → q) ↔ (~ p∨ ~ q)

Negation for Implication

~ (p → q) ↔ ( p∧ ~ q)

Biconditional Sentences (p ↔ q)↔(p → q)∧(q → p)+ (p↔q)↔(p ∧ q) ∨ (~ p∧ ~ q)]