f What is the key feature of the Venn diagram from Part E?
A
a 1. If an integer is divisible by 10, then its last digit is 0. 2. If the last digit of an integer is 0, then it is divisible by 10.
B
b
C
c
D
d
E
e It must be one circle only.
F
f See solution.
Practice makes perfect
a Let's analyze the given biconditional!
An integer is divisible by $10$ if and only if its last digit is $0$
We can write the two conditionals by taking the first part as a hypothesis and the second one as conclusion, and then reverse it. This gives us the following conditional statements.
If an integer is divisible by10, then its last digit is 0.
If the last digit of an integer is 0, then it is divisible by 10.
b Let's review what a Venn diagram is! It illustrates how the set of things that satisfy the hypothesis lies inside the set of things that satisfy the conclusion. In the first conditional, the hypothesis is that an integer is divisible by 10, and the conclusion is that the integer's last digit is 0. It gives us the following Venn diagram.
c In this case, the hypothesis is that an integer is divisible by 10, and the conclusion is that the integer's last digit is 0. This gives us the following Venn diagram.
d As we can tell from Part A and Part B, the set of integers divisible by 10 is a part of the set of integers with 0 as the last digit is a part of the set of integers divisible by 10. Therefore, these sets are the same set!
e As we can tell from Part D, a Venn diagram for a true biconditional is only one circle. If it was not the case, then one of the conditionals would not be true. Thus, the Venn diagram must be one circle for a biconditional to be true.
f From Part E we know that a Venn diagram for a true biconditional is only one circle. Thus, both sides p and q in a biconditional p ↔ q represent the same set. From this we can see that we can write a good definition as a biconditional, because the definition of any term must represent exactly the same term.
