Question

• $subseteq$ means \is a subset of.\ • $subset$ means \is a proper subset of.\ • $

subseteq$ means \is not a subset of.\ • $varnothing$ is the empty set. for each statement, decide if it is true or false. (a) ${1,2,3,4,5}subseteq{1,3}$ true false (b) ${12,14}
subseteq{11,12,13,14,15,ldots}$ true false (c) ${22,23,24,26}subset{22,23,24,26}$ true false (d) $varnothingsubseteq{c,g,j,m}$ true false

Explanation:

Step1: Recall subset definition

A set $A$ is a subset of set $B$ ($A\subseteq B$) if every element of $A$ is in $B$. For (a), in $\{1,2,3,4,5\}\subseteq\{1,3\}$, elements $2,4,5$ of the first - set are not in the second set. So it's false.

Step2: Recall non - subset definition

A set $A$ is not a subset of set $B$ ($A
subseteq B$) if there is at least one element of $A$ that is not in $B$. For (b), in $\{12,14\}
subseteq\{11,12,13,14,15,\ldots\}$, since $12$ and $14$ are in the second set, it's false.

Step3: Recall proper - subset definition

A set $A$ is a proper subset of set $B$ ($A\subset B$) if $A$ is a subset of $B$ and $A
eq B$. For (c), $\{22,23,24,26\}$ and $\{22,23,24,26\}$ are equal sets, so $\{22,23,24,26\}\subset\{22,23,24,26\}$ is false.

Step4: Recall empty - set property

The empty set $\varnothing$ is a subset of every set. For (d), $\varnothing\subseteq\{c,g,j,m\}$ is true.

Answer:

(a) False
(b) False
(c) False
(d) True