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Logic Worksheet

Logic Worksheet
  • Page 1
 1.  
The basic axiom of algebra represented by (5 + 8) + 10 = 5 + (8 + 10) is ____
a.
associative property of addition
b.
distributive property of addition
c.
commutative property of addition
d.
commutative property of multiplication


Solution:

(5 + 8) + 10 = 5 + (8 + 10)

It is in the form of (a + b) + c = a + (b + c), where a, b and c are any real numbers.

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


Correct answer : (1)
 2.  
The method of assuming the opposite of a statement is true. If this leads to impossibility, then the original statement is true is:
a.
A proof by contradiction
b.
A Direct proof


Solution:

In an indirect proof, or a proof by contradiction, you assume the opposite of a statement is true. If this leads to impossibility, then the original statement is true.


Correct answer : (1)
 3.  
The rule in mathematics that we accept to be true without proof is called:
a.
An axiom
b.
A conjecture
c.
A theorem


Solution:

An axiom is a statement which is accepted to be true without proof.


Correct answer : (1)
 4.  
A statement that is believed to be true but not yet proved is:
a.
A theorem
b.
An axiom
c.
A conjecture


Solution:

A conjecture is a statement that is believed to be true but not yet proved.


Correct answer : (3)
 5.  
The basic axiom of algebra represented by 10 + (- 10) = 0 is
a.
Closure property of addition
b.
Identity property of addition
c.
Inverse property of addition
d.
Inverse property of multiplication


Solution:

10 + (- 10) = 0.

It is of the form a + (- a) = 0, where a is any real number.

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


Correct answer : (3)
 6.  
The basic axiom of algebra represented by 4 · (14) = 1, is
a.
Inverse property of multiplication
b.
Distributive property of multiplication
c.
Identity property of multiplication
d.
Associative property of multiplication


Solution:

4 · (14) = 1

It is of the form a · (1a) = 1, where a is any real number not equal to zero.

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


Correct answer : (1)
 7.  
The statement: "There is no rational number whose square is 2", is:
a.
False
b.
True


Solution:

Assume that there exists a rational number of the form ab, where a and b are integers that have no common factors other than 1 and b ≠ 0, whose square is 2.

So, (ab)2 = 2.

a2b2 = 2

a2 = 2b2
[Multiply each side by b2.]

This implies that 2 is a factor of a2. Therefore, 2 is also a factor of a. Thus a can be written as 2n.

(2n)2 = 2b2
[Replace a with 2n in step 4.]

4n2 = 2b2
[Simplify.]

2n2 = b2
[Divide each side by 2.]

This implies that 2 is a factor of b2 and also a factor of b.

So, 2 is a factor of both a and b.

This is impossible because a and b have no common factors other than 1.

Therefore, it is impossible that there exists a rational number whose square is 2.

The statement: "There is no rational number whose square is 2", is true.


Correct answer : (2)
 8.  
The statement: "If a and b are both even integers, then their sum is even", is:
a.
True
b.
False


Solution:

Assume that a + b is odd.

Then a + b = 2n + 1.
[Assume the opposite of a + b even is true.]

a = 2n + 1 - b
[Subtract b from each side.]

a = 2n + 1 – 2m
[b is even: b = 2m.]

a = 2(n – m) + 1

This implies that a is odd, which contradicts the given statement that a is an even integer.

Therefore, it is impossible that a + b is odd .

So, a + b must be even if a and b are even.


Correct answer : (1)
 9.  
The basic axiom of algebra represented by a × 1 = a where a is any real number, is:
a.
Associative property of multiplication
b.
Inverse property of multiplication
c.
Identity property of multiplication
d.
Commutative property of multiplication


Solution:

a × 1 = a, where a is any real number.

The basic axiom of algebra represents the above, is identity property of multiplication.


Correct answer : (3)
 10.  
The basic axiom of algebra represented by 2t + 7 = 7 + 2t, where t is any real number, is
a.
Commutative property of addition
b.
Associative property of addition
c.
Inverse property of addition
d.
Identity property of addition


Solution:

2t + 7 = 7 + 2t, where t is any real number.

It is of the form a + b = b + a.

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


Correct answer : (1)

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