Answer:
4. Conditional: If two angles form a linear pair, then they are supplementary.
Inverse: If two angles do not form a linear pair, then they are not supplementary.
Converse: If two angles are supplementary, then they form a linear pair.
truth value — conditional: true; inverse: false; converse: false.
5. Conditional: If a capital letter is a vowel, then it is symmetrical.
Converse: If a capital letter is symmetrical, then it is a vowel.
truth: biconditional — false.
6. Conditional: If a number is divisible by 6, then it is divisible by 3.
Converse: If a number is divisible by 3, then it is divisible by 6.
truth: biconditional — false.
Explanation:
The problem text says "inverse" when it probably means "converse." Since we're not totally sure of the intent, we have provided both the inverse and converse here. The Converse blank should be filled with the converse.
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The patterns you're trying to fill in are ...
Conditional: If p, then q.
Inverse: If not p, then not q.
Converse: If q, then p.
For problems 5 and 6, you're given a biconditional. Its form is ...
p if and only if q.
That is, you can identify the clauses p and q from this form and use them in the forms of the conditional and converse, as required.
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4. You're given the wording for p and q. All you have to do is insert those words into the form for a conditional:
Conditional: If two angles form a linear pair, then they are supplementary.
For the converse, you need to adjust the wording to conform to standard English. That is, "they" can replace "two angles" and vice versa.
Converse: If two angles are supplementary, then they form a linear pair.
In order to determine the truth value, you need to refer to your knowledge of geometry. A linear pair of angles is always supplementary. (That's part of the definition of a linear pair.) Thus the conditional is true. However, angles may be supplementary without being part of a linear pair. Hence both the inverse and converse are false.
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5. When you extract the p and q clauses from the biconditional, you have ...
p: a capital letter is a vowel
q: it is symmetrical
These get filled into the forms for a conditional and its converse to give ...
Conditional: If a capital letter is a vowel, then it is symmetrical.
As in problem 4, the nouns and pronouns need to be swapped for the converse.
Converse: If a capital letter is symmetrical, then it is a vowel.
When you consider the truth value, you need to consult your knowledge of the alphabet and the classifications of the letters. Here's my read on the truth:
- conditional — truth depends on the meaning of "symmetrical". AIOUY have horizontal symmetry; E has vertical symmetry, but not horizontal symmetry. If E is considered "symmetrical", then the conditional is true.
- converse — false. (H, M, T, V, W, X have horizontal symmetry, are not vowels)
A biconditional is only true if both the conditional and converse are true. They are not, so the biconditional is false.
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6. When you extract the p and q clauses from the biconditional, you have ...
p: a number is divisible by 6
q: it is divisible by 3
These get filled into the forms for a conditional and its converse to give ...
Conditional: If a number is divisible by 6, then it is divisible by 3.
As in problem 4, the nouns and pronouns need to be swapped for the converse.
Converse: If a number is divisible by 3, then it is divisible by 6.
When you consider the truth value, you need to consult your knowledge of number divisibility. A number divisible by 6 is both even and divisible by 3. An odd number divisible by 3 will not be divisible by 6. So, the conditional is true and its converse is false. That means the biconditional is false.