Sure, let's analyze each part of the question step by step:
Given sets:
[tex]$
\begin{array}{l}
P=\{2,4,6,8, \ldots\} \quad \text{(Even numbers starting from 2)} \\
S=\{3,6,9,12, \ldots\} \quad \text{(Multiples of 3 starting from 3)} \\
T=\{6,12,18,24, \ldots\} \quad \text{(Multiples of 6 starting from 6)}
\end{array}
$[/tex]
a) Is [tex]\(6\)[/tex] in [tex]\(P\)[/tex], [tex]\(S\)[/tex], and [tex]\(T\)[/tex] simultaneously?
We need to check if 6 is an element of all three sets.
- 6 is present in [tex]\( P \)[/tex] since it is an even number.
- 6 is present in [tex]\( S \)[/tex] since it is a multiple of 3.
- 6 is present in [tex]\( T \)[/tex] since it is a multiple of 6.
Since 6 is in all three sets, the statement is True.
b) Is [tex]\( P \subseteq T \)[/tex] or [tex]\( S \subseteq T \)[/tex]?
We need to check if all elements of [tex]\( P \)[/tex] are in [tex]\( T \)[/tex] or if all elements of [tex]\( S \)[/tex] are in [tex]\( T \)[/tex].
- [tex]\( P \)[/tex] (even numbers) includes numbers like 2, 4, which are not multiples of 6, so [tex]\( P \subseteq T \)[/tex] is not true.
- [tex]\( S \)[/tex] (multiples of 3) includes numbers like 3, 9, which are not multiples of 6, so [tex]\( S \subseteq T \)[/tex] is not true.
Since neither [tex]\( P \subseteq T \)[/tex] nor [tex]\( S \subseteq T \)[/tex] is true, the statement is False.
c) Is [tex]\( T \subseteq P \)[/tex] and [tex]\( T \subseteq S \)[/tex]?
We need to check if all elements of [tex]\( T \)[/tex] are in [tex]\( P \)[/tex] and also in [tex]\( S \)[/tex]:
- All elements in [tex]\( T \)[/tex] (multiples of 6) are even numbers, so [tex]\( T \subseteq P \)[/tex].
- All elements in [tex]\( T \)[/tex] (multiples of 6) are also multiples of 3, so [tex]\( T \subseteq S \)[/tex].
Since both [tex]\( T \subseteq P \)[/tex] and [tex]\( T \subseteq S \)[/tex] are true, the statement is True.
d) Is the intersection of [tex]\( P \)[/tex], [tex]\( S \)[/tex], and [tex]\( T \)[/tex] non-empty?
We need to check if there is at least one common element in sets [tex]\( P \)[/tex], [tex]\( S \)[/tex], and [tex]\( T \)[/tex]:
- The number 6 is in [tex]\( P \)[/tex], [tex]\( S \)[/tex], and [tex]\( T \)[/tex].
Since the intersection is non-empty, the statement is True.
So the outcomes are:
- a) True
- b) False
- c) True
- d) True
