a If a rock is a certain type, then it was created in a certain way. Apply this to the three different kinds of rock.
B
b The converse of a conditional statement, q→ p, exchanges the hypothesis and the conclusion of the conditional statement.
C
c The inverse of a conditional statement, ~ p→ ~ q, requires us to negate the hypothesis and the conclusion of the conditional statement.
A
aIgneous: If a rock is igneous, then it is formed from the cooling of molten rock.
Sedimentary: If a rock is sedimentary, then it is formed from pieces of other rock. Metamorphic: If a rock is metamorphic, then it is formed by changing temperature, pressure, or chemistry.
B
bIgneous: If a rock is formed from the cooling of molten rock, then it is igneous. This is a true statement, see solution.
Sedimentary: If a rock is formed from pieces of other rock, then it is sedimentary. This is a true statement, see solution. Metamorphic: If a rock is formed by changing temperature, pressure, or chemistry, then it is metamorphic. This is a true statement, see solution.
C
cExample Solution: If a rock is not sedimentary, then it is not formed from pieces of other rock.
Converse: If a rock is not formed from pieces of other rock, then it is not sedimentary. This is a true statement, see solution.
Practice makes perfect
a We have three kinds of rocks that form under different circumstances. According to the exercise, igneous rock is formed by cooling molten rock. Let's write this in if-then form.
If a rock is igneous,
then it is formed from the
cooling of molten rock.We also know that sedimentary rock is formed from pieces of other rock, which can be written as an if-then statement.
If a rock is sedimentary,
then it is formed from pieces of other rock.
Finally, we have been told that metamorphic rock is formed by changing temperature, pressure, or chemistry. Let's write this in if-then form as well.
If a rock is metamorphic,
then it is formed by changing temperature,
pressure, or chemistry.
b The converse of a conditional statement, q→ p, exchanges the hypothesis and the conclusion of the conditional statement. Let's do this for each statement in Part A and then asses if they are true or not.
If a rock is formed from the
cooling of molten rock,
then it is igneous.Since all rocks formed from cooling molten rock are called igneous, this is a true statement.
If a rock is formed from pieces of other rock,
then it is sedimentary.
Since all rocks formed from pieces of other rock are called sedimentary, this is a true statement.
If a rock is formed by changing
temperature, pressure, or chemistry,
then it is metamorphic.
Since all rocks formed by changing temperature, pressure, or chemistry are metamorphic, this is also a true statement.
c We can, for example, write the inverse of any of the three statements we wrote in Part A. The inverse of a conditional statement, ~ p→ ~ q, requires us to negate the hypothesis and the conclusion of the conditional statement.
If a rock is not sedimentary,
then it is not formed from pieces of other rock.
The converse of the inverse will be the contrapositive of the original statement.
If a rock is not formed by changing
temperature, pressure, or chemistry,
then it is not metamorphic.
Since the contrapositive and conditional statement are logically equivalent, and the conditional statement is true, we know that the contrapositive will also be true.
