﻿ Logic Worksheet | Problems & Solutions

# 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.

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.

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.

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.

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.

6.
The basic axiom of algebra represented by 4 · ($\frac{1}{4}$) = 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.

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.

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 Ã¢â‚¬â€œ 2m
[b is even: b = 2m.]

a = 2(n Ã¢â‚¬â€œ 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.

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.

10.
The basic axiom of algebra represented by 2$t$ + 7 = 7 + 2$t$, where $t$ is any real number, is

#### 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.