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Math 144 Preview Activities 3.3

Raymond Juan Sanchez

April 13, 2015

T.A: Kaylee Hamann

Section 3.3 Preview Activity 1:

1. Use truth tables to explain why  P   ∨ ∼ P   is a tautology and  P   ∧ ∼ P   is a contradiction.

P   P   ∨ ∼ P P   ∧ ∼ P 

T T FF T FBy definition, we can see that the first statement is a tautology and the other statement is a contra-

diction.

2. Use a truth table to show that  ∼  (P   ⇒ Q) is a logically equivalent to  P   ∧ ∼ Q.

P Q   ∼ (P   ⇒ Q)   P   ∧ ∼ QT T F FT F T TF T F F

F F F FFrom the truth table we can deduce that the two statements are logically equivalent.

3. Give a counterexample to show that the following statement is false

For each real number x1

x(1 − x) ≥ 4

A counterexample is to let  x = 2 then we get

1

2(1 − 2) =

  1

−2 

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4. Given the new statement: For each real number x, if 0  < x  0,  y > 0, and

x

y +

 y

x ≤ 2

1. Algebraically manipulate this inequality

x

y +

 y

x  ≤   2

x2 + y2

xy  ≤   2

x2 + y2 ≤   2xy

x2 + y2 − 2xy   ≤   0

(x − y)2 ≤   0

3. Explain why the last inequality from part (2) leads to a contradiction.

It leads to a contradiction because the quantity (x − y)2 is always positive. This inequality would holdfor x and y if  x =  y , then (x − y)2 = 0, but we’ve assumed that  x = y , hence the last inequality is alwaysfalse given that we’ve assumed  x = y.

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