Logic and Propositions

proposition/statement

a sentence that can be true or false BUT not both

Is the following text a statement?

9 + 10 = 21

Statement

Mathematical equations are statements

Is the following text a statement?

I like pizza

Statement

Opinions are statements

Is the following a statement?

Is 2 + 6 = 8?

Not a statement

Negation definition

Negates true/ false of a propositon

How is ū read as

u-bar or negation of u

Create the truth table of ū given u:

U - T, F

Ū - F, T

Conjunction definition

combining two propositions as a composition

What is the conjunction of “X and Y”

X ∧ Y

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, F, F

Rewrite the following sentence logically:

“T but not U”

T ∧ Ū

disjunction definition

finding where one, the other, or both of a composition is true

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

X - T, T, F, F

Y - T, F, T, F

X ∨ Y - T, T, T, F

Prove that (Ā ∨ Ē) is the negation of (A ∧ E) given A and E:

A - T, T, F, F

E - T, F, T, F

A ∧ E - T, F, F, F

Ā - F, F, T, T

Ē - F, T, F, T

Ā ∨ Ē - F, T, T, T

Prove that (Ā ∧ Ē) is the negation of (A ∨ E) given A and E:

A - T, T, F, F

E - T, F, T, F

A ∨ E - T, T, T, F

Ā - F, F, T, T

Ē - F, T, F, T

Ā ∧ Ē - F, F, F, T

List two ways (X ∨ Y) can be elaborated

• (X ∧ Y)

• (T/F)

Rewrite the following function to reflect it’s precedence:

Ā ∧ Ē ∨ Ī ∧ Ō

(Ā ∧ Ē) ∨ (Ī ∧ Ō)

Conditional proposition/implication

Forms an “if-then” statement between propositions

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