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A relation R:A→AR:A→A is said to be reflexive if, for every element aa in the set AA (where AA is a non-empty set), the ordered pair (a,a)(a,a) belongs to the relation RR. In simpler terms, reflexive relations include every element paired with itself in the set.

To prove that the function f:R→Rf:R→R given by f(x)=2xf(x)=2x is one-to-one (injective), we need to show that if f(x1)=f(x2)f(x1)=f(x2), then x1=x2x1=x2 for all x1,x2x1,x2 in the domain.

Let's assume f(x1)=f(x2)f(x1)=f(x2): 2x1=2x22x1=2x2

Now, we'll solve for x1x1 and x2x2: x1=x2x1=x2

Since x1=x2x1=x2, it means that for any two inputs x1x1 and x2x2 that produce the same output under the function f(x)=2xf(x)=2x, those inputs must be the same. This proves that the function f(x)=2xf(x)=2x is one-to-one.

To prove that (f+g)∘h=f∘h+g∘h(f+g)∘h=f∘h+g∘h, let's start by understanding what (f+g)∘h(f+g)∘h means:

(f+g)∘h(x)=(f+g)(h(x))=f(h(x))+g(h(x))(f+g)∘h(x)=(f+g)(h(x))=f(h(x))+g(h(x))

Now, let's find (f∘h+g∘h)(x)(f∘h+g∘h)(x):

f∘h(x)+g∘h(x)=f(h(x))+g(h(x))f∘h(x)+g∘h(x)=f(h(x))+g(h(x))

This expression is identical to what we found for (f+g)∘h(x)(f+g)∘h(x). Hence, we can conclude that (f+g)∘h=f∘h+g∘h(f+g)∘h=f∘h+g∘h.

To find (g∘f)(x)(g∘f)(x), which is the composition of g(x)g(x) with f(x)f(x), we substitute f(x)f(x) into g(x)g(x) wherever we see xx.

Given:

f(x)=∣x∣f(x)=∣x∣ g(x)=∣5x+1∣g(x)=∣5x+1∣

We first find f(x)f(x):

f(x)=∣x∣f(x)=∣x∣

And then substitute it into g(x)g(x):

g(f(x))=∣5(∣x∣)+1∣g(f(x))=∣5(∣x∣)+1∣

Now, ∣x∣∣x∣ can be either xx if x≥0x≥0 or −x−x if x<0x<0.

So, ∣5(∣x∣)+1∣∣5(∣x∣)+1∣ will be:

If x≥0x≥0: g(f(x))=∣5x+1∣g(f(x))=∣5x+1∣

If x<0x<0: g(f(x))=∣−5x+1∣g(f(x))=∣−5x+1∣

In H3PO2, also known as hypophosphorous acid, the oxidation number of hydrogen (H) is typically +1.

The sum of the oxidation numbers in a neutral molecule must equal zero. Since there are three hydrogen atoms, each with an oxidation number of +1, their total contribution is +3.

For oxygen (O), the typical oxidation number is -2, except in peroxides and when it's bonded to fluorine. In H3PO2, oxygen's oxidation number is -1.

Given that the overall charge of the molecule is zero, and knowing the oxidation numbers of hydrogen and oxygen, you can calculate the oxidation number of phosphorus (P).

Let's denote the oxidation number of phosphorus as xx:

(+1×3)+(−1×2)+x=0(+1×3)+(−1×2)+x=0

3−2+x=03−2+x=0

1+x=01+x=0

x=−1x=−1

So, in H3PO2, the oxidation number of phosphorus is -1.

The increase in density from titanium (Z = 22) to copper (Z = 29) in the first series of transition elements can be explained by several factors:

1. Atomic Mass: As you move from titanium to copper in the periodic table, the atomic mass generally increases due to the addition of more protons, neutrons, and electrons. Since density is mass per unit volume, an increase in atomic mass tends to increase density.

2. Atomic Radius: While the atomic radius generally decreases across a period in the periodic table due to increasing effective nuclear charge, the increase in atomic mass across the transition metals offsets this effect to some extent. As you move from titanium to copper, the increase in atomic mass generally outweighs the decrease in atomic radius, contributing to the increase in density.

3. Crystal Structure: Transition metals typically have a close-packed crystal structure, which means that their atoms are densely packed together in a regular pattern. Changes in atomic size and mass can influence how tightly packed these atoms are, affecting the density of the material.

4. Electron Configuration: Transition metals have complex electron configurations, with electrons occupying different sublevels within the d-block. Changes in electron configuration can influence the interactions between atoms and hence affect the density of the material.

5. Transition Metals' Special Properties: Transition metals often exhibit unique properties such as high melting points, hardness, and metallic bonding characteristics, all of which can influence the density of the elements in this series.

Overall, the increase in density from titanium to copper in the first series of transition elements is a result of various interplaying factors including atomic mass, atomic radius, crystal structure, electron configuration, and special properties of transition metals.

When undecomposed silver bromide (AgBr) is washed with hypo solution (sodium thiosulfate) in photography, it forms a complex ion known as the tetrathionate complex, [Ag(S2O3)2]3-. This complex ion helps in removing the unexposed silver bromide from the photographic film during the fixing process, leaving behind the developed silver image.