# Deductive Reasoning Worksheet

Deductive Reasoning Worksheet
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1.
The basic axiom of algebra represented by 5 + (- 5) = 0 is:
 a. Inverse property of addition b. Closure property of addition c. Identity property of addition d. Inverse property of multiplication

#### Solution:

5 + (- 5) = 0.

It is of the form x + (- x) = 0, where Ã¢â‚¬ËœxÃ¢â‚¬â„¢ is any real number.

So, the basic axiom of algebra represents the above is inverse property of addition.

2.
The basic axiom of algebra represented by 9 · ($\frac{1}{9}$) = 1, is
 a. Inverse property of multiplication b. Identity property of multiplication c. Associative property of multiplication d. Distributive property of multiplication

#### Solution:

9 · (19) = 1

It is of the form a · (1a) = 1, where Ã¢â‚¬ËœaÃ¢â‚¬â„¢ is any real number not equal to zero.

So, the basic axiom of algebra represents the above is inverse property of multiplication.

3.
Select the correct statement / statements.
(i). A convincing argument that uses a deductive reasoning is called a proof.
(ii). A conjucture that is proven is a theorem.
(iii). A convincing argument that uses a deductive reasoning is called a postulate.
 a. (ii) only b. (ii) and (iii) only c. (i) only d. (i) and (ii) only

#### Solution:

A convincing argument that uses a deductive reasoning is called a proof.

A conjucture that is proven is a theorem.

Statements (i) and (ii) are only correct.

4.
Select the correct statement(s).
I. Deductive reasoning arrives at a valid conclusion by logical reasoning based on the available facts.
II. Deductive reasoning is the only method to prove theorems.
III. In geometry, postulates and properties are accepted as true and these are used in deductive reasoning to prove other statements.
 a. I, and III b. I, II, and III c. II only d. I only

#### Solution:

Deductive reasoning is a process of reasoning logically from given facts to a conclusion.

There are methods other than deductive reasoning in geometry to prove theorems.

Postulates and Properties are accepted as true in geometry. Deductive reasoning uses these postulates and properties.

5.
The statement, " The product of two odd integers is odd", is:
 a. True b. False

#### Solution:

Let a and b be two odd integers.

Then a = 2 n + 1 and b = 2 m + 1.

ab = (2 n + 1) (2 m + 1)

ab = 4 nm + 2 n + 2 m + 1
[Simplify.]

ab = 2 (2 nm + n + m) + 1

So, ab is an odd integer.

Therefore, the product of two odd integers is odd is true.

6.
The statement, " The sum of an odd integer and an even integer is an odd integer", is:
 a. True b. False

#### Solution:

Let a be an odd integer and b be an even integer.

Then a = 2 n + 1 and b = 2 m.

a + b = (2 n + 1) + 2 m

a + b = 2 n + 2 m + 1

a + b = 2 (n + m) + 1

This implies that a + b is odd.

Therefore, the sum of an odd and even integers is an odd integer is true.

7.
The statement, "The difference of a number and its reversal of a two digit number is a multiple of 9" is:
 a. True b. False

#### Solution:

Let the units place be a and tens place be b in the two digit number.

Then the number = 10b + a.

Reversed number = 10a + b.

(10 b + a) - (10a + b)
[Difference between the number and its reverse.]

= 10b + a - 10a - b

= 9b - 9a

= 9 (b - a)

This implies that the difference of a two digit number and its reversal is a multiple of 9 is true.

8.
The statement, " the sum of a two digit number and its reversal is a multiple of 11 ", is :
 a. True b. False

#### Solution:

Let the units place be a and tens place be b in the two digit number.

Then the number = 10b + a.

Reversed number = 10a + b.

(10b + a) + (10a + b)
[Sum of a number and its reverse.]

= 11b + 11a
[Simplify.]

= 11(b + a)

This implies that the sum of a two digit number and its reversal is a multiple of 11 is true.

9.
The statement, "The square of an even integer is even", is:
 a. False b. True

#### Solution:

Let a be an even integer.

Then a = 2 n

a2 = (2 n)2
[Square each side of the equation.]

a2 = 4n2
[Simplify.]

a2 = 2 (2n2)

This implies that a2 is even.

Therefore, the statement "the square of an even integer is even", is true.

10.
The statement, " If the square of an integer is even, then the integer is even", is:
 a. False b. True

#### Solution:

Let a2 be an even integer and assume that a is odd.

Then a = 2 n + 1

a2 = (2 n + 1)2
[Square each side of the equation.]

a2 = 4 n2 + 4 n + 1
[Use formula (x + y)2 = x2 + 2xy + y2.]

a2 = 2 (2 n2 + 2 n) + 1

This implies that a2 is odd, which contradicts our assumption that a2 is an even integer.

So, a is an even integer.

Therefore, the statement "if the square of an integer is even, then the integer is even ", is true.