Finding relation between elements

Question

I have seen the following type of problems in reasoning tests:

If $A \le B = C > F < G = L$ , then which of the following is true?

a. $A < G$ and $A >F$

b. $C >F$

c. $G =B$

[Answers are given by me just as examples.]

Can anybody explain a nice general method of solving these problems?

Answer 1

I assume that $A,B,C,F,G,L$ are real numbers. I think it is better to rewrite the assumption this way :

$\begin{eqnarray} A & \leq & B \tag{1}\\ B & = & C \tag{2}\\ F & < & C \tag{3}\\ F & < & G \tag{4}\\ G & = & L \tag{5}\\ \end{eqnarray} $

Using $(2)$, inequalities $(3)$ and $(4)$ give $F < B$ and $F < G$. So, we know that $B$ and $G$ are both greater than $F$ but not necessarily equal. (For example, take $A=0$, $B=C=1$, $F=0$ and $G=L=2$. In this example, we don't have $G=B$ but $(1)$ to $(5)$ hold.

From $(3)$, the statment $C>F$ is obviously true.

The statment $A<G$ is false. Consider $A=-1$, $B=C=2$, $G=L=1$ and $F=0$. $(1)$ to $(5)$ still hold but we don't have $A<G$. The statment $A > F$ is also false.

To find such counter-examples, you could draw a line (which represents $\mathbb{R}$) and place $A,B,C,F,G,L$ on this line according to $(1)$,...,$(5)$. It might also help you to see which statement is true or not.

Answer 2

Here's how I would do it. This isn't 100% maximally efficient, but it will get you the correct answer consistently.

Firstly, gather equal terms.

$$A \leq \{B,C\} > F < \{L,G\}$$

Then reorganize the available information.

1. $A \leq \{B,C\}$
2. $F < \{B,C\}$
3. $F < \{L,G\}$

Then reorganize the available answers.

a. $A < G$ and $F < A$

b. $F < C$

c. $G = B$

Now just read off the answer. In this case, we can see that 2 implies b. (Thus b is the correct answer).

Its also important to remember that if $x < y$, then $x \leq y.$

This kind of reasoning works in quasi-ordered sets, but note that more sophisticated arguments are valid in partially ordered sets, and especially in a totally ordered sets.

Answer 3

I would usually start by cancelling out answers that are definitely not right... But it's obvious it's b because it's in the question -_-

Answer 4

One can apply the principle:

No conclusion can be drawn about the ordering of items separated by at least one $\lt$ or $\leqslant$ and at least one $\gt$ or $\geqslant$.

In the present case, this means that A can only be compared to B and C, hence anything comparing A to F, G or L fails. Thus, 1. fails. Likewise, G can only be compared to L and F, hence anything comparing G to A, B or C fails. Thus, 3. fails.

One is left with 2., which is a part of the hypothesis.