Identify the statements that are true. (a) The expression always gives odd numbers. (b) All even numbers can be expressed as . (c) Both expressions and describe all odd numbers. (d) The expression gives both even and odd numbers.

We will check each statement to determine if it is true or false. Step 1 — Checking statement (a) Let be any integer. We know that is an even number. This is because can be written as . An even number minus 1 is always an odd number. So, will always be an odd number. Let us try some values for . If : If : If : All these results are odd numbers. > Step 2 — Checking statement (b) Let be any integer. First, let us check if always gives an even number. The term is an even number. The number 4 is also an even number. An even number minus an even number is always an even number. So, always gives even numbers. Now, let us check if it gives all even numbers. Let us try to get the even number . We set the expression equal to . The value is not an integer. So, cannot be expressed in this form. Not all even numbers can be expressed as . > Step 3 — Checking statement (c) Let be any integer. The expression represents an even number () plus 1. An even number plus 1 is always an odd number. This is the standard way to write any odd number. For example, if , . If , . If , . Let be any integer. The expression represents an even number () minus 1. An even number minus 1 is always an odd number. This form also represents any odd number. For example, if , . If , . If , . Both expressions generate the set of all odd integers. > Step 4 — Checking statement (d) Let be any integer. The term is always an even number. The number 3 is an odd number. An even number plus an odd number is always an odd number. So, will always give odd numbers. Let us try some values for . If : If : If : All these results are odd numbers. The expression only gives odd numbers. > Answer > (a) True > (b) False > (c) True > (d) False

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Question 25

Identify the statements that are true.

(a) The expression $4m - 1$ always gives odd numbers. (b) All even numbers can be expressed as $6j - 4$ . (c) Both expressions $2p + 1$ and $2q - 1$ describe all odd numbers. (d) The expression $2f + 3$ gives both even and odd numbers.

Solution

We will check each statement to determine if it is true or false.

Step 1 — Checking statement (a)

Let $m$ be any integer. We know that $4m$ is an even number. This is because $4m$ can be written as $2 \times (2m)$ . An even number minus 1 is always an odd number. So, $4m - 1$ will always be an odd number. Let us try some values for $m$ . If $m = 1$ : $4(1) - 1$ $= 4 - 1$ $= \mathbf{3}$ If $m = 2$ : $4(2) - 1$ $= 8 - 1$ $= \mathbf{7}$ If $m = 0$ : $4(0) - 1$ $= 0 - 1$ $= \mathbf{-1}$ All these results are odd numbers.

$\boxed{\text{Statement (a) is TRUE}}$

Step 2 — Checking statement (b)

Let $j$ be any integer. First, let us check if $6j - 4$ always gives an even number. The term $6j$ is an even number. The number 4 is also an even number. An even number minus an even number is always an even number. So, $6j - 4$ always gives even numbers. Now, let us check if it gives all even numbers. Let us try to get the even number $\mathbf{0}$ . We set the expression equal to $\mathbf{0}$ . $6j - 4 = 0$ $6j = 4$ $j = \frac{4}{6}$ $j = \frac{2}{3}$ The value $j = \frac{2}{3}$ is not an integer. So, $0$ cannot be expressed in this form. Not all even numbers can be expressed as $6j - 4$ .

$\boxed{\text{Statement (b) is FALSE}}$

Step 3 — Checking statement (c)

Let $p$ be any integer. The expression $2p + 1$ represents an even number ( $2p$ ) plus 1. An even number plus 1 is always an odd number. This is the standard way to write any odd number. For example, if $p = 0$ , $2(0) + 1 = 1$ . If $p = 1$ , $2(1) + 1 = 3$ . If $p = -1$ , $2(-1) + 1 = -1$ . Let $q$ be any integer. The expression $2q - 1$ represents an even number ( $2q$ ) minus 1. An even number minus 1 is always an odd number. This form also represents any odd number. For example, if $q = 1$ , $2(1) - 1 = 1$ . If $q = 2$ , $2(2) - 1 = 3$ . If $q = 0$ , $2(0) - 1 = -1$ . Both expressions generate the set of all odd integers.

$\boxed{\text{Statement (c) is TRUE}}$

Step 4 — Checking statement (d)

Let $f$ be any integer. The term $2f$ is always an even number. The number 3 is an odd number. An even number plus an odd number is always an odd number. So, $2f + 3$ will always give odd numbers. Let us try some values for $f$ . If $f = 1$ : $2(1) + 3$ $= 2 + 3$ $= \mathbf{5}$ If $f = 2$ : $2(2) + 3$ $= 4 + 3$ $= \mathbf{7}$ If $f = 0$ : $2(0) + 3$ $= 0 + 3$ $= \mathbf{3}$ All these results are odd numbers. The expression $2f + 3$ only gives odd numbers.

$\boxed{\text{Statement (d) is FALSE}}$

Answer

(a) True (b) False (c) True (d) False

Question 25

Answer

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