Determine whether the following biconditional statements are true or false.
1. Any whole number is divisible by 3 if and only if the sum of its digits is divisible by 3.
2. 1/x > 0 if and only if x ≥ 0.
3. 2 is a prime if and only if 7 is even.
#1 read about casting out threes and casting out nines
#2 false. 1/0 is not defined
#3 since 7 is not even, what do you think?
To determine whether the biconditional statements are true or false, we need to evaluate both directions of the statements separately.
1. Any whole number is divisible by 3 if and only if the sum of its digits is divisible by 3.
To determine the truth of this statement, let's examine both directions:
- If a whole number is divisible by 3, then the sum of its digits is divisible by 3:
To check if this is true, we can look at some examples. For instance, if we take the number 15, which is divisible by 3, the sum of its digits is 1 + 5 = 6, which is also divisible by 3. Similarly, if we take the number 27, the sum of its digits is 2 + 7 = 9, another multiple of 3. By testing various numbers, we can see that when a number is divisible by 3, the sum of its digits is also divisible by 3. So, this direction holds true.
- If the sum of the digits of a whole number is divisible by 3, then the number itself is divisible by 3:
Let's test this direction using counterexamples. If we take the number 14, the sum of its digits is 1 + 4 = 5, which is not divisible by 3. Similarly, if we take the number 29, the sum of its digits is 2 + 9 = 11, which is also not divisible by 3. Therefore, this direction does not hold true.
Since one direction of the biconditional statement is true, and the other is false, we can conclude that the first statement is a false biconditional statement.
2. 1/x > 0 if and only if x ≥ 0.
To determine the truth of this biconditional statement, let's examine both directions:
- If 1/x > 0, then x ≥ 0:
To check if this holds true, we need to consider different cases. If we select a positive value for x, such as x = 2, then 1/2 > 0, which is true. Similarly, if we take x = 0, the inequality becomes undefined. However, if we pick a negative value for x, such as x = -2, then 1/(-2) > 0 becomes -1/2 > 0, which is not true. Based on these examples, we can conclude that this direction holds true.
- If x ≥ 0, then 1/x > 0:
Examining this direction, we can see that it is always true. When x is equal to or greater than zero, 1/x will always yield a positive value.
Since both directions of the biconditional statement are true, we can conclude that the second statement is a true biconditional statement.
3. 2 is a prime if and only if 7 is even.
Let's evaluate both directions of this biconditional statement:
- If 2 is a prime, then 7 is even:
This direction holds false since 7 is an odd number, not an even number.
- If 7 is even, then 2 is a prime:
This direction is also false since 2 is indeed a prime number, but 7 is not even.
As both directions of the biconditional statement are false, we can conclude that the third statement is a false biconditional statement.