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In24-S1-MA1014 - Mathematics

In24-S1-MA1014 - Mathematics

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Which of the following are True about Real Numbers $\mathbb{R}$ \mathbb{R}?

$\forall a \in \mathbb{R} \exists b \in \mathbb{R}; a>b$ \forall a \in \mathbb{R} \exists b \in \mathbb{R}; a>b

✅

$\forall a\in \mathbb{R}-\{0\}; a>0\Leftrightarrow a^{-1}>0$ \forall a\in \mathbb{R}-\{0\}; a>0\Leftrightarrow a^{-1}>0

✅

$\exists b \in \mathbb{R} \forall a \in \mathbb{R}; a>b$ \exists b \in \mathbb{R} \forall a \in \mathbb{R}; a>b

❌

$\exists a \in \mathbb{R} \exists b \in \mathbb{R}; a>b$ \exists a \in \mathbb{R} \exists b \in \mathbb{R}; a>b

✅

$\forall a \in \mathbb{R} \forall b \in \mathbb{R}; a>b$ \forall a \in \mathbb{R} \forall b \in \mathbb{R}; a>b

❌

$\exists a,b,c \in \mathbb{R}; a+(b\cdot c)=(a+b)\cdot (a+c)$ \exists a,b,c \in \mathbb{R}; a+(b\cdot c)=(a+b)\cdot (a+c)

✅

$\forall a,b,c \in \mathbb{R}; a\cdot (b+c)=(a\cdot b)+(a \cdot c)$ \forall a,b,c \in \mathbb{R}; a\cdot (b+c)=(a\cdot b)+(a \cdot c)

✅

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Let A be a non empty subset of $\mathbb{R}$ \mathbb{R} which is bounded below. What is True about $\inf A$ \inf A?

It always exists in $\mathbb{R}$ \mathbb{R}

✅

It is equal to $\min A$ \min A

❌

It always exists in $\mathbb{Z}$ \mathbb{Z} if A is a non empty subset of $\mathbb{Z}$ \mathbb{Z} which is bounded below.

❌

It is a lower bound

✅

It is the supremum of the set of upper bounds of A

❌

It is the infimum of the set of lower bounds of A

❌

It is equal to $-\sup(-A)$ -\sup(-A) where $-A=\{-x | x\in A\}$ -A=\{-x | x\in A\}

✅

It is the minimum of the set of lower bounds of A

❌

It is the maximum of the set of upper bounds of A

❌

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