Question

List the triples in the relation\(\{ (a,b,c)|a,b\;{\bf{and}}\;\;c\,{\bf{are}}{\rm{ }}{\bf{integers}}{\rm{ }}{\bf{with}}\;0 < a < b < c < 5\} \).

Short answer

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

Step-by-step solution

Given data

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

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\(A\).

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: \(R = \{ (1,2,3),(1,2,4),(1,3,4),(2,3,4)\} \).