Conditional Propositions

Conditional proposition/implication

Forms an “if-then” statement between propositions

Rewrite the following statement symbolically:

“If x, then y”

x → y

Identify the hypothesis and conclusion of the following conditional proposition:

x → y

x is the hypothesis

y is the conclusion

Create the truth table of X → Y given X and Y:

X - T, T, F, F

Y - T, F, T, F

X → Y - T, F, T, T

Paying for expedited shipping but receiving a package late is an example of what conditional proposition

Given X → Y

X is true, Y is false

Getting two candy bars from a vending machine while only paying for one is an example of what conditional proposition?

Given X → Y

X is false, Y is true

Find the contrapositive of Ā → E

Ē → A

T/F - “If it it rains, then I do not have an umbrella” and “If I have an umbrella, it does not rain” have the same meaning

True

Contrapositives have the same meaning

≡ meaning

logically equivalent

Find the negation of Ū → Ē

Ū → E

Find the converse of Ō → A

A → Ō

Find the inverse of Ī → Ē

I → E

Write the inverse of the following statement:

“I’ll call you if you are not home”

If you are home, I will not call you

Write the converse of the following statement:

“The weather is raining implies that the weather is pouring”

The weather pouring implies that the weather is raining

Find the contrapositive of the following statement:

“A relationship is healthy unless there are not boundaries”

If a relationship is not healthy, there are no boundaries (double negative)

Biconditional proposition

Forms an “if-and-only-if” statement between propositions

T/F - The following statements yield the same truth:

“The triangles are congruent only if their sides are the same length”

“The triangles are congruent if and only if their sides are the same length”

False

First statement conditional, second

Rewrite the following statement symbolically:

“x if and only if y”

x ↔ y

Create the truth table for x ↔ y given the following:

x: T, T, F, F

y: T, F, T, F

x ↔ y: T, F, F, T

A ∨ E ≡ ?

• Ā → E

• Ē → A

Equivalent disjunction

OR

tautology

propositions that are always true

contradiction

propositions that are always false

negating a contradiction will always yield a _____

tautology

contingency

a proposition that could be both false or true depending on truth values

Ā ∨ Ē ∨ A is a _____

tautology

(A ∨ E) → (A ∧ E) is a _____

tautology

(A ∧ E) → (Ã ∨ Ē) is a ____

tautology

Ā ∧ E ∧ Ō ∧ A is a ____

contradiction

(A ∧ E) ∧ (Ã ∨ Ē) is a ____

contradiction