Determine if a true bioconditional can be written from each conditional statement. If not, gove a counterexample.
1. If a = b then |a| = |b|
Is this true?
2. If y^2= 64 then 3y=24
Is this false? Because if you do -8 both of them aren't equal
#1 of course not: |-6|=|6| but -6≠6
#2 same reasoning. y could be -8.
You can say, therefore that if |a|=|b|, then a^2 = b^2.
1. Is true
To determine if a true biconditional can be written from a conditional statement, we need to check if the converse of the conditional statement is also true.
A conditional statement takes the form "If A, then B". The converse of this statement is "If B, then A". A true biconditional can be written if and only if both the conditional statement and its converse are true.
1. Conditional statement: If a = b, then |a| = |b|
To find the converse, we switch the positions of "a = b" and "|a| = |b|".
Converse: If |a| = |b|, then a = b
Now, let's evaluate the conditional statement and its converse:
- Conditional statement: If a = b, then |a| = |b|
This statement is true because if a and b are equal, the absolute values of a and b will also be equal.
- Converse: If |a| = |b|, then a = b
This statement is also true because if the absolute values of a and b are equal, it implies that a and b have the same value.
Since both the conditional statement and its converse are true, a true biconditional can be written based on this conditional statement.
2. Conditional statement: If y^2 = 64, then 3y = 24
To find the converse, we switch the positions of "y^2 = 64" and "3y = 24".
Converse: If 3y = 24, then y^2 = 64
Now, let's evaluate the conditional statement and its converse:
- Conditional statement: If y^2 = 64, then 3y = 24
This statement is false because there are other possible values for y that satisfy y^2 = 64. For example, y could be -8, which would make 3y = -24.
- Converse: If 3y = 24, then y^2 = 64
This statement is also false because if 3y = 24, it implies that y = 8, which would make y^2 = 64. However, there can be other values for y (like -8) that make 3y = 24 but do not satisfy y^2 = 64.
Therefore, a true biconditional cannot be written based on this conditional statement, and a counterexample can be provided by setting y = -8.