Verified answer

Question:-

Determine wheather the below relation is reflexive, symmetric and transitive: Relation R on the set N natural numbers is defined as R = {(x, y) : y = x + 5 and x < 4}.

Given:-

Relation R on the set N natural numbers is defined as R = {(x, y) : y = x + 5 and x < 4}.

To Find:-

Determine wheather the below relation is reflexive, symmetric and transitive.

Solution:-

It is given that Relation R in the set N of natural numbers defined as R = {(x, y) : y = x + 5 and x < 4}.

Clearly, R = {(1, 6), (2, 7), (3, 8)}.

● Reflexive:-

A relation is said to be reflexive if (x, x) ∈ R, ∀x ∈ N.

We can see that (1, 1) ∉ R.

⇒ R is not Reflexive.

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● Symmetric:-

A relation is said to be symmetric if (y, x) ∈ R whenever (x, y) ∈ R.

Here, (1, 6) ∈ R, but (6, 1) ∉ R.

⇒ R is not Symmetric.

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● Transitive:-

Now, since there is no pair in R such that (x, y) and

(y, z) ∈ R, then (x, z) belong to R.

⇒ R is not Transitive.

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Answer:-

Therefore, R is not reflexive, nor symmetric, not transitive.

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Hope you have satisfied. ⚘

Answer:

Question:-

• ᴅᴇᴛᴇʀᴍɪɴᴇ ᴡʜᴇᴀᴛʜᴇʀ ᴛʜᴇ ʙᴇʟᴏᴡ ʀᴇʟᴀᴛɪᴏɴ ɪꜱ ʀᴇꜰʟᴇxɪᴠᴇ, ꜱʏᴍᴍᴇᴛʀɪᴄ ᴀɴᴅ ᴛʀᴀɴꜱɪᴛɪᴠᴇ: ʀᴇʟᴀᴛɪᴏɴ ʀ ᴏɴ ᴛʜᴇ ꜱᴇᴛ ɴ ɴᴀᴛᴜʀᴀʟ ɴᴜᴍʙᴇʀꜱ ɪꜱ ᴅᴇꜰɪɴᴇᴅ ᴀꜱ ʀ = {(x, ʏ) : ʏ = x + 5 ᴀɴᴅ x < 4}.

Solution:-

ʀ={(x,ʏ):ʏ=x+5ᴀɴᴅx<4}={(1,6),(2,7),(3,8)}

ɪᴛ ɪꜱ ꜱᴇᴇɴ ᴛʜᴀᴛ (1,1)∈/ ʀ⇒ʀ ɪꜱ ɴᴏᴛ ʀᴇꜰʟᴇxɪᴠᴇ. ᴀʟꜱᴏ (1,6)∈ʀ.

ʙᴜᴛ, (1,6)∈/ ʀ. ∴ʀ ɪꜱ ɴᴏᴛ ꜱʏᴍᴍᴇᴛʀɪᴄ.

ɴᴏᴡ, ꜱɪɴᴄᴇ ᴛʜᴇʀᴇ ɪꜱ ɴᴏ ᴘᴀɪʀ ɪɴ ʀ ꜱᴜᴄʜ ᴛʜᴀᴛ (x,ʏ) ᴀɴᴅ (ʏ,ᴢ)∈ʀ, ᴛʜᴇɴ (x,ᴢ) ᴄᴀɴɴᴏᴛ ʙᴇʟᴏɴɢ ᴛᴏ ʀ.

∴ ʀ ɪꜱ ᴛʀᴀɴꜱɪᴛɪᴠᴇ.

ʜᴇɴᴄᴇ, ʀ ɪꜱ ɴᴇɪᴛʜᴇʀ ʀᴇꜰʟᴇxɪᴠᴇ, ɴᴏʀ ꜱʏᴍᴍᴇᴛʀɪᴄ, ʙᴜᴛ ᴛʀᴀɴꜱɪᴛɪᴠᴇ.

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