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write the following boolean expressions in an equivalent sum of product canonical form in three variables x1, x2, and x3:

1. x1*x2 ?

3. (x1+X2)'*X3

1. x1*x2 ?

3. (x1+X2)'*X3

Let D = {1, 2, 3}. The domain of the variables x and y will be D. Is it possible to find a predicate P(x, y) such that ∃y ∀x P(x, y) is true but ∀x ∃y P(x, y) is false? Explain

Determine if the following argument is valid using truth tables.

p ∧ q → r

__________

∴ q → r

p ∧ q → r

__________

∴ q → r

Let D = {1, 2, 3}. The domain of the variables x and y will be D.

Give an example of a predicate P(x, y) such that ∀x ∃y P(x, y) is true, but ∃y ∀x P(x, y) is false

Give an example of a predicate P(x, y) such that ∀x ∃y P(x, y) is true, but ∃y ∀x P(x, y) is false

Determine if the following argument is valid using truth tables.

p → (q ∨ r)

∼ q

__________

∴ p → r

p → (q ∨ r)

∼ q

__________

∴ p → r

Determine if the following argument is valid using truth tables

p → q

p → r

__________

∴ p → (q ∧ r)

p → q

p → r

__________

∴ p → (q ∧ r)

Determine if the following argument is valid using truth tables

p ←→ q

p∧ ∼ q

_________

∴ r

p ←→ q

p∧ ∼ q

_________

∴ r

Determine if the following argument is valid using truth tables.

p −→ r

r

_______

∴ p

p −→ r

r

_______

∴ p

Determine if the following argument is valid using truth tables.

p ∧ q → r

__________

∴ q → r

p ∧ q → r

__________

∴ q → r