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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
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
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
Determine if the following argument is valid using truth tables.

p → (q ∨ r)
∼ q
__________
∴ p → r
Determine if the following argument is valid using truth tables

p → q
p → r
__________
∴ p → (q ∧ r)
Determine if the following argument is valid using truth tables

p ←→ q
p∧ ∼ q
_________
∴ r
Determine if the following argument is valid using truth tables.


p −→ r
r
_______
∴ p
Determine if the following argument is valid using truth tables.

p ∧ q → r
__________
∴ q → r
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
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