Implication and Equivalence

We have now seen three logical operators in propositoinal logic: conjunction (∧), disjunction (∨), and negation (¬). In this section we will look at two more: implication (⇒), and equivalence (⇔).

Eight-minute video

You can also view this video on YouTube

You can find the slides here and also as .odp.

Key Points

Implication

Implication roughly means “if p then q”.

When the left-hand side of the implication is false, then the result is always true. Thus false ⇒ true is true. This is counterintuitive. When a statement like this is found to be true, it is said to be vacuously true.

Truth table for implication

p q p⇒q
T T T
T F F
F T T
F F T

Equivalence

Equivalence, or the biconditional, roughly means “if and only if”.

Two propositions p and q are equivalent if and only if they both have the same truth value.

Truth table for equivalence

p q p⇔q
T T T
T F F
F T F
F F T

Questions

1. Check your understanding

1. Implication

Calculate the truth values for the following implications:

  Expression True False  
1.
2.
3.
4.

Check Answers

2. Equivalence

Work out the truth value for the following equivalences:

  Expression True False  
1.
2.
3.
4.

Check Answers

3. Vacuous Truth

Which of the following statements are vacuously true?

Check Answers

4. Tautology and Contradiction

Which of the following statements are tautologies and which are contradictions?

  Expression Tautology Contradiction  
1. p ⇔ ¬p
2. p ⇒ p ∨ q
3. p ∧ ¬p
4. p ⇔ (p ⇒ false)
5. (p ⇒ q) ⇔ (¬p ∨ q)

Check Answers

Summary

In this section we have learned two more logical operators: implication (⇒), and equivalence (⇔).

  • You should be able to use and construct the truth tables for implication and equivalence.
  • You should know what it means for a proposition to be tautological or a contradiction.
  • You should know what it means for a proposition to be vacuously true.

In the next section we will find out about bitwise operators.