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Expert-verified Found in: Page 589 ### Discrete Mathematics and its Applications

Book edition 7th
Author(s) Kenneth H. Rosen
Pages 808 pages
ISBN 9780073383095 # List the triples in the relation$$\{ (a,b,c)|a,b\;{\bf{and}}\;\;c\,{\bf{are}}{\rm{ }}{\bf{integers}}{\rm{ }}{\bf{with}}\;0 < a < b < c < 5\}$$.

The resultant answer is $$\{ (1,2,3),(1,2,4),(1,3,4),(2,3,4)\}$$.

See the step by step solution

## Step 1: Given data

$$R = \{ (a,b,c)\mid a,b,c$$ are integers with $$0 < a < b < c < 5\}$$.

## Step 2: Concept ofsets

The concept of set is a very basic one. It is simple; yet, it suffices as the basis on which all abstract notions in mathematics can be built. $$A$$ set is determined by its elements. If $$A$$ is a set, write $$x \in A$$ to say that $$x$$ is an element of $$A$$.

## Step 3: Simplify the expression

This $$0 < a < b < c < 5$$ implies that $$a$$ can only take on the value 1 or 2 as $$b$$ and $$c$$ need to be larger than $$a$$ while also smaller than 5. When $$a = 1$$, then $$b$$ can be 2 or 3 as $$b$$ needs to be larger than $$a$$ and $$c$$ need to be larger than $$b$$ while also smaller than 5.

When $$a = 1$$ and $$b = 2$$, then $$c$$ can take on the value 3 or 4 (as $$c$$ needs to be smaller 5 and larger than $$b$$ )

$$\begin{array}{l}(1,2,3) \in R\\(1,2,4) \in R\end{array}$$

When $$a = 1$$ and $$b = 3$$, then $$c$$ can only take on the value of 4 (as $$c$$ needs to be smaller 5 and larger than $$b$$ ) $$(1,3,4) \in R$$

When $$a = 2$$, then $$b = 3$$ and $$c = 4$$ (as $$c$$ needs to be smaller 5 and larger than $$b$$, while $$b$$ is larger than $$a$$ ) $$(2,3,4) \in R$$

The relation $$R$$ is then the set of all found triples: $$R = \{ (1,2,3),(1,2,4),(1,3,4),(2,3,4)\}$$. ### Want to see more solutions like these? 